2. Sub-ballistic regime

In this section we classify limits of the RTRW model where the embedded random walk has a positive bias and the holding times have infinite mean. The main aim is to prove Theorem 1.1, which we do by considering the Laplace transform arguments of [Reference Černý and Wassmer18] and applying topological results of [Reference Whitt44]. These results of [Reference Whitt44] will play an important role, and we also direct the reader to [Reference Ben Arous, Cabezas, Černý and Royfman3, Reference Ben Arous and Černỳ5, Reference Černý and Wassmer18, Reference Meerschaert and Scheffler39, Reference Mourrat40, Reference Zindy45] for more information and similar usage. For $x \in {\mathbb{Z}}$ recall that

\begin{equation*}\varphi_{\omega_x}(\lambda)=\int e^{-\lambda u}\omega_x({\textrm{d}} u)={\textit{E}}^\omega[e^{-\lambda \eta_{x,1}}]\end{equation*}

is the quenched Laplace transform of $\omega_x$. Recall from (1.1) that $\upsilon_\beta=(\beta-1)/(\beta+1)$ is the speed of the embedded walk Y. By the gambler’s ruin this is the probability that Y never returns to the origin. For convenience we write ${\mathbb{P}}^Y(\!\cdot\!)\,{:\!=}\,{\mathbb{P}}(\cdot|Y)$ and ${\textit{P}}^{\omega,Y}(\!\cdot\!)\,{:\!=}\,{\textit{P}}^\omega(\cdot|Y)$. We begin, in Proposition 2.1, by showing that the one-dimensional distributions of the scaled clock process converge.

Proposition 2.1. Under the assumptions of Theorem 1.1, for any $\lambda>0$ and $t>0$,

(2.1) \begin{equation}\lim_{k\rightarrow \infty}{\mathbb{E}}\left[\exp\left(-\lambda \frac{S_{n_kt}}{a_{n_k}}\right)\right]=\exp\Bigg(-\upsilon_\beta^2 t\sum_{\ell=1}^\infty f(\ell,\lambda)(1-\upsilon_\beta)^{\ell-1}\Bigg).\end{equation}

Proof. For simplicity we write n for $n_k$, with the understanding that we consider subsequences along which (1.3) holds. Conditional on Y, the holding times at different vertices are independent; therefore,

\begin{align*} {\mathbb{E}}^Y\left[\exp\left(-\lambda \frac{S_{nt}}{a_n}\right)\right] &= {\mathbb{E}}^Y\left[\exp\left(-\frac{\lambda}{a_n}\sum_{x \in {\mathbb{Z}}}\sum_{i=1}^{{\mathcal{L}}(x,\lfloor nt\rfloor-1)}\eta_{x,i}\right)\right] \\ &= \prod_{x \in {\mathbb{Z}}} {\mathbb{E}}^Y\left[\exp\left(-\frac{\lambda}{a_n}\sum_{i=1}^{{\mathcal{L}}(x,\lfloor nt\rfloor-1)}\eta_{x,i}\right)\right]\!. \end{align*}

Under ${\textit{P}}^{\omega,Y}$, the holding times at a given vertex are independent and do not depend on Y; therefore we can write the above expression as

\begin{align*} \prod_{x \in {\mathbb{Z}}} {\textbf{E}}\left[{\textit{E}}^{\omega,Y}\left[\prod_{i=1}^{{\mathcal{L}}(x,\lfloor nt\rfloor-1)}e^{-\frac{\lambda}{a_n}\eta_{x,i}}\right]\right] &= \prod_{x \in {\mathbb{Z}}}{\textbf{E}}\left[\varphi_{\omega_x}\left(\frac{\lambda}{a_n}\right)^{{\mathcal{L}}(x,\lfloor nt\rfloor -1)}\right] \\ &= \prod_{\ell=1}^\infty\prod_{x \in {\mathcal{R}}_\ell(nt)}{\textbf{E}}\left[\varphi_{\omega_x}\left(\frac{\lambda}{a_n}\right)^\ell\right]\!,\end{align*}

where ${\mathcal{R}}_\ell(m)=\{x \in {\mathbb{Z}}\ : \ {\mathcal{L}}(x,\lfloor m\rfloor-1)=\ell\}$ is the number of vertices with local time $\ell$ at time $m-1$. By translation-invariance we then have that

(2.2) \begin{align} {\mathbb{E}}^Y \left[\exp\left(-\lambda \frac{S_{nt}}{a_n}\right)\right] &= \prod_{\ell=1}^\infty{\textbf{E}} \left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]^{|{\mathcal{R}}_\ell(nt)|} \nonumber \\ &= \exp\left(\sum_{\ell=1}^\infty |{\mathcal{R}}_\ell(nt)|\log\left({\textbf{E}} \left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right)\right)\!.\end{align}

Write ${\mathcal{R}}_\ell^+(m)=\{x \in {\mathbb{Z}}\ : \ {\mathcal{L}}(x,\lfloor m\rfloor-1)\geq \ell\}$; then, by the main result of [Reference Pitt42], for fixed $t>0$, $|{\mathcal{R}}_\ell(nt)|/\lfloor nt\rfloor$ converges a.s. to $\upsilon_\beta^2(1-\upsilon_\beta)^{\ell-1}$, and $|{\mathcal{R}}_\ell^+(nt)|/\lfloor nt\rfloor$ converges a.s. to $\upsilon_\beta(1-\upsilon_\beta)^{\ell-1}$ as ${n \rightarrow \infty}$. The local time at the origin stochastically dominates that of any other vertex; therefore,

\begin{equation*} \frac{{\textit{E}}[|{\mathcal{R}}_\ell^+(m)|]}{m} = \frac{1}{m}\sum_{x=-m}^m {\textit{P}}({\mathcal{L}}(x,m)\geq \ell) \; \leq \; \frac{2m +1}{m}{\textit{P}}({\mathcal{L}}(0,m)\geq \ell).\end{equation*}

The local time ${\mathcal{L}}(0,m)$ is increasing in m, so for each $\ell\geq 1$ we have that ${\textit{P}}({\mathcal{L}}(0,m)\geq \ell)\leq$ $\leq{\textit{P}}({\mathcal{L}}(0,\infty)\geq \ell)= (1-\upsilon_\beta)^{\ell-1}$, since ${\mathcal{L}}(0,\infty)$ is geometrically distributed with success probability $\upsilon_\beta$. It follows that

(2.3) \begin{equation}\frac{{\textit{E}}[|{\mathcal{R}}_\ell^+(nt)|]}{nt} \leq C(1-\upsilon_\beta)^\ell\end{equation}

uniformly over $n,t,\ell$ such that $nt\geq 1$. For fixed $M \in {\mathbb{N}}$, by using the convergence of $|{\mathcal{R}}_\ell(nt)|/\lfloor nt\rfloor$ and the assumptions on $\varphi_{\omega_0}$ (from (1.3)), we have that for ${\textit{P}}$-almost every Y,

(2.4) \begin{equation} \sum_{\ell=1}^M |{\mathcal{R}}_\ell(nt)|\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right) \rightarrow -\upsilon_\beta^2t\sum_{\ell=1}^Mf(\ell,\lambda)(1-\upsilon_\beta)^{\ell-1}.\end{equation}

By Jensen’s inequality and (1.3),

\begin{equation*}-\log({\textbf{E}}[\varphi_{\omega_0}(\lambda/a_n)^\ell])\;\leq\; -\ell\log({\textbf{E}}[\varphi_{\omega_0}(\lambda/a_n)]) \;\leq\; C\ell\lambda^\alpha n^{-1};\end{equation*}

thus, f grows at most linearly in $\ell$, and therefore we have that $\sum_{\ell=1}^Mf(\ell,\lambda)(1-\upsilon_\beta)^{\ell-1}$ converges as $M\rightarrow \infty$. Moreover,

\begin{equation*} -\sum_{\ell=M}^\infty |{\mathcal{R}}_\ell(nt)|\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right) \;\leq\; \frac{C\lambda^\alpha}{n}\sum_{\ell=M}^\infty \ell|{\mathcal{R}}_\ell(nt)|.\end{equation*}

By Markov’s inequality and (2.3) we have that for $c_1 \in (0,-\log(1-\upsilon_\beta))$ and some $\tilde{c}>0$,

\begin{equation*}{\textit{P}}\left(\frac{|{\mathcal{R}}_\ell^+(nt)|}{nt}\geq e^{-c_1\ell}\right)\;\leq\; {\textit{E}}\left[\frac{|{\mathcal{R}}_\ell^+(nt)|}{nt}\right]e^{c_1\ell} \;\leq\; C(1-\upsilon_\beta)^\ell e^{c_1\ell} \;\leq\; Ce^{-\tilde{c}\ell}.\end{equation*}

A union bound then shows that

\begin{equation*}{\textit{P}}\left(\bigcup_{\ell\geq M} \left\{\frac{|{\mathcal{R}}_\ell(nt)|}{nt}\geq e^{-c_1\ell}\right\}\right) \leq C\sum_{\ell\geq M}e^{-\tilde{c}\ell} \leq Ce^{-\tilde{c}M}.\end{equation*}

It then follows that, for $\delta>0$,

\begin{equation*} {\textit{P}}\left( -\sum_{\ell=M}^\infty |{\mathcal{R}}_\ell(nt)|\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right)\geq \delta\right) \;\leq\; Ce^{-\tilde{c}M}+{\textbf{1}}_{\left\{Ct\lambda^\alpha\sum_{\ell=M}^\infty \ell e^{-c_1\ell}\geq \delta\right\}},\end{equation*}

which converges to 0 as $M\rightarrow \infty$ independently of n. In particular, combining with (2.4),

\begin{equation*} \sum_{\ell=1}^\infty |{\mathcal{R}}_\ell(nt)|\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right) \rightarrow -\upsilon_\beta^2 t\sum_{\ell=1}^\infty f(\ell,\lambda)(1-\upsilon_\beta)^{\ell-1}\end{equation*}

in ${\textit{P}}$-probability; therefore, recalling (2.2), by bounded convergence we have that (2.1) holds.□

Before proving the main result we state a technical lemma which allows us to compare the process with a version in which we resample the environment at fixed times. This will allow us to show that the limiting process has independent increments. The result follows similarly to [Reference Černý and Wassmer18, Lemma 3.2] for the d-dimensional RTRW, so we omit the proof.

Fix $d\in {\mathbb{N}}$, $0<t_1<\ldots<t_d=t$, and let $\overline{\omega}\,{:\!=}\,(\omega_x^j)_{x \in {\mathbb{Z}}, j=1,\ldots,d}$ be a sequence of independent and identically distributed (i.i.d.) $\pi$-random measures. For $x \in {\mathbb{Z}}$, $j=1,\ldots,d$, and $i\geq 1$, let $\eta_{x,i}^j$ be independent with law $\omega_x^j$. Let j(x) be such that $nt_{j(x)-1}\leq \Delta^Y_x< nt_{j(x)}$; that is, j(x) denotes the index of the interval in which x is first reached. Define

\begin{equation*}S'_{\!\!m}\,{:\!=}\sum_{x \in {\mathbb{Z}}}\sum_{i=1}^{{\mathcal{L}}(x,m-1)}\eta_{x,i}^{j(x)};\end{equation*}

then $$S'\mathop = \limits^d S$$ with respect to the annealed law ${\mathbb{P}}$. The clock process $S'_{\!\!m}$ can be thought of as the sum of the first m holding times where we refresh the entire unseen environment at times $\lfloor nt_j\rfloor$. We then define the approximation of S $\prime$:

\begin{equation*}S''_{\!\!\!\!m}=\sum_{j=1}^d\sum_{k=\lfloor nt_{j-1}\rfloor}^{(m\land \lfloor nt_j\rfloor)-1} \eta_{Y_k,{\mathcal{L}}(Y_k,k)}^j.\end{equation*}

The process $S''_{\!\!\!\!m}$ can be thought of as the sum of the first m holding times where we refresh the entire environment at times $\lfloor nt_j\rfloor$. By independence of the environment and the walk between times $\lfloor nt_{j-1}\rfloor$ and $\lfloor nt_j\rfloor$, we have that the differences

\begin{equation*}S''_{\lfloor nt_j\rfloor}-S''_{\lfloor nt_{j-1}\rfloor}=\sum_{k=\lfloor nt_{j-1}\rfloor}^{\lfloor nt_j\rfloor-1} \eta_{Y_k,{\mathcal{L}}(Y_k,k)}^j\end{equation*}

are independent.

Lemma 2.1. Under the assumptions of Theorem 1.1 we have that for any $\varepsilon>0$,

\begin{equation*}{\mathbb{P}}\left(\sup_{s \leq t}|S''_{\lfloor ns\rfloor}-S'_{\lfloor ns\rfloor}|>\varepsilon a_n\right)\end{equation*}

converges to 0 as ${n \rightarrow \infty}$.

Using these two results we can conclude the proof of Theorem 1.1 by applying results of [Reference Whitt44].

Proof of Theorem 1.1. We begin by completing the result for the clock process $S_t$. By Proposition 2.1 we have that the one-dimensional distributions of $S_{n_kt}/a_{n_k}$ converge to those of a process ${\mathcal{S}}_t$ whose Laplace transform is given by the right-hand side of (2.1). With a slight abuse of notation, we continue to use ${\mathbb{P}}$ for the continuum law. It is immediate from the Laplace transform and the fact that $f(\ell,\lambda)\leq C\ell\lambda$ that the limit ${\mathcal{S}}_t$ is ${\mathbb{P}}$-a.s. finite for any $t>0$. Moreover, ${\mathbb{P}}$-a.s. we have that ${\mathcal{S}}_0=0$, since

(2.5) \begin{equation}\lim_{t \rightarrow 0}{\mathbb{E}}[\exp(\!-\lambda{\mathcal{S}}_t)] =1.\end{equation}

A consequence of Lemma 2.1, similarly to [Reference Černý and Wassmer18, Lemma 3.2], is that the laws of $S_{n_kt}/a_{n_k}$ converge in the sense of finite-dimensional distributions to those of ${\mathcal{S}}_t$, and the limit process has independent, non-negative increments. It then follows from equivalent characterisations of convergence for monotone functions [Reference Whitt44, Theorem 13.6.3] and the fact that $S_{n_kt}$ in increasing in t that $S_{n_kt}/a_{n_k}$ converges in distribution to the process ${\mathcal{S}}_t$ with respect to the $M_1$ topology.

The clock process $S_m$ has stationary increments because the traps in the environment are i.i.d. and the transition probabilities of the embedded walk are translation-invariant. Using the Laplace transform and convergence of the finite-dimensional distributions, we therefore have that ${\mathcal{S}}$ also has stationary increments. To show that ${\mathcal{S}}$ is a Lévy process, it remains to show that for every $\varepsilon>0$ and $t\geq 0$ we have that $\lim_{h\rightarrow 0^+}{\mathbb{P}}(|{\mathcal{S}}_{t+h}-{\mathcal{S}}_t|>\varepsilon)=0$. This follows from (2.5) since ${\mathcal{S}}$ has stationary and independent increments.

The limit process ${\mathcal{S}}_t$ belongs to $D([0,\infty),{\mathbb{R}})$ and is unbounded and strictly increasing. By continuity of the inverse at strictly increasing functions [Reference Whitt44, Corollary 13.6.4], the inverse map between unbounded, increasing functions in $D_{M_1}$ onto $D_{U}$ is continuous at unbounded, strictly increasing functions. Therefore, the sequence $S^{-1}_{a_{n_k}t}/n_k$ converges in ${\mathbb{P}}$-distribution on $D_{U}$ to ${\mathcal{S}}^{-1}_t$.

The main result then follows from continuity of composition at continuous limits [Reference Whitt44, Theorem 13.2.1], since $Y_{nt}/n$ converges ${\mathbb{P}}$-a.s. on $D_{U}([0,\infty),{\mathbb{R}})$ to the process $t\upsilon_\beta$, and

\[X_{a_{n_k}t}/n_k=Y_{S^{-1}_{a_{n_k}t}}/n_k.\]

We want to show that if the convergence in (1.3) holds along all subsequences then $f(\ell,\lambda)=g(\ell)\lambda^\alpha$ for a function $g\not\equiv0$. Notice that

(2.6) \begin{equation}-n\log\!\Bigg({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_{n}}\right)^\ell\right]\Bigg)=-n\log\!\Bigg({\textbf{E}}\Bigg[{\textit{E}}^\omega\bigg[\exp\bigg(-\frac{1}{(n\lambda^{-\alpha})^{1/\alpha}L(n\lambda^{-\alpha})}\frac{L(n\lambda^{-\alpha})}{L(n)}\eta_0\bigg)\bigg]^\ell\Bigg]\Bigg).\end{equation}

Since L is slowly varying, for $\lambda>0$ fixed we have that $L(n\lambda^{-\alpha})/L(n)$ converges to 1 as ${n \rightarrow \infty}$. In particular, using computations similar to those of Proposition 2.1, it is straightforward to show that, for each $\ell\in{\mathbb{N}}$, the right-hand side of (2.6) converges to $f(\ell,1)\lambda^\alpha$. By assumption we have that the left-hand side of (2.6) converges to $f(\ell,\lambda)$; therefore we indeed have that $f(\ell,\lambda)=g(\ell)\lambda^\alpha$ for $g(\ell)=f(\ell,1)$.

To show that ${\mathcal{S}}_t$ is an $\alpha$-stable subordinator it suffices to note that ${\mathcal{S}}_t$ is increasing and self-similar:

$${S_t}\; = \;\mathop {\lim }\limits_{n \to \infty } {{{S_{nt}}} \over {{a_n}}}\; = \;\mathop {\lim }\limits_{n \to \infty } {{{a_{\lambda n}}} \over {{a_n}}} \cdot {{{S_{\lambda n{t \over \lambda }}}} \over {{a_{\lambda n}}}}\;\mathop = \limits^{\rm{d}} \;{S_{t/\lambda }}\mathop {\lim }\limits_{n \to \infty } {{{a_{\lambda n}}} \over {{a_n}}}\; = \;{\lambda ^{1/\alpha }}{S_{t/\lambda }}.$$

□

Fix $\mu\in(0,1)$ and $\gamma>\mu^{-1}$; then $\alpha\,{:\!=}\log(\mu^{-1})/\log(\gamma)\in(0,1)$. Let $(\kappa_x)_{x\in{\mathbb{Z}}}$ be an i.i.d. sequence of geometrically distributed random variables with success probability $1-\mu$ under ${\mathbb{P}}$; that is, ${\mathbb{P}}(\kappa_x=k)=(1-\mu)\mu^k$ for $k\geq 0$. We then define the holding times by ${\textit{P}}^\omega(\eta_{x,j}=\gamma^{\kappa_x})=1$.

For $\ell\in{\mathbb{N}}$ and $\lambda>0$ we have that

(2.7) \begin{equation}-n\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(-\frac{\lambda}{a_n}\right)^\ell\right]\right) = -n\log\left({\textbf{E}}\left[\exp\left(-\frac{\ell\lambda}{a_n}\gamma^{\kappa_0}\right)\right]\right)\!,\end{equation}

which converges to a non-trivial limit for some $a_n$ if and only if $\gamma^{\kappa_0}$ belongs to the domain of attraction of a stable law. However,

(2.8) \begin{equation}n{\mathbb{P}}(\gamma^{\kappa_0}\geq ta_n)=n{\mathbb{P}}\left(\kappa_0\geq \frac{\log(ta_n)}{\log(\gamma)}\right)=n\mu^{\left\lceil\frac{\log(ta_n)}{\log(\gamma)}\right\rceil},\end{equation}

which only converges along certain subsequences with $a_n=n^{1/\alpha}$ and does not converge in general for any suitable $a_n$. Therefore, $\gamma^{\kappa_0}$ does not belong to the domain of attraction of any stable law, and (2.7) does not converge to a function f satisfying the requirements of Theorem 1.1.

Furthermore, noting also that

\begin{align*}{\textbf{E}}\left[\exp\left(-\frac{\ell\lambda}{a_n}\gamma^{\kappa_0}\right)\right]&=\int_0^1{\mathbb{P}}\left(\exp\left(\frac{-\ell\lambda}{a_n}\gamma^\kappa_0\right)\geq t\right){\textrm{d}} t \\&= 1-\ell\lambda\int_0^\infty\exp\left(-\ell\lambda t\right){\mathbb{P}}(\gamma^\kappa_0>ta_n){\textrm{d}} t,\end{align*}

where

\begin{equation*}\mu t^{-\alpha}\leq n{\mathbb{P}}(\gamma^{\kappa_0}>tn^{1/\alpha})\leq t^{-\alpha},\end{equation*}

we see that (2.7) converges along subsequences $n_k$ such that (2.8) converges with $a_n=n^{1/\alpha}$, and the limiting function satisfies the conditions of Theorem 1.1.

3. Ballistic regime

In this section we consider the regime in which ${\mathbb{E}}[\eta_0]<\infty$ but ${\mathbb{E}}[\eta_0^2]=\infty$. In this setting we do not have a central limit theorem for X; however, by [Reference Bowditch17, Corollary 2.2] we have that if $\beta>1$ then $X_{nt}n^{-1}$ converges ${\mathbb{P}}$-a.s. to $\nu_\beta t$, where $\nu_\beta$ is given in (1.2). Recall that we define the first hitting times $\Delta_m\,{:\!=}\inf\{t\geq 0\ :\ X_t=m\}$. The main aim of the section is to prove Theorem 1.2, which shows that $(\Delta_{nt\nu_\beta}-nt)/a_n$ and $(X_{nt}-nt\nu_\beta)/a_n$ converge jointly to a pair of Lévy processes under certain assumptions on the distribution of the holding times and for a suitable scaling $a_n$.

Our strategy will be as follows. We begin by using a regeneration structure in order to approximate $\Delta_{nt}$ by a sum of i.i.d. random variables. We then prove convergence of the Laplace transforms of these variables. This completes the results for $\Delta_{nt}$, which we then extend to the inverse by using standard inverse results of [Reference Whitt44] for processes with linear centring. We then conclude the result for $X_{nt}$ by showing that it does not deviate too far from an appropriate transformation of the inverse of $\Delta_{nt}$.

We begin by exploiting the renewal structure of the walk. Let $\zeta^Y_0=0$ and, for $j=1,2,\ldots$, define

\[\zeta^Y_j\,{:\!=}\inf\big\{m>\zeta^Y_{j-1}\ : \ \{Y_r\}_{r=0}^{m-1} \cap \{Y_r\}_{r=m}^\infty = \emptyset \big\}\]

to be the regeneration times of the walk Y. We then have that $\zeta^X_j\,{:\!=}\,S_{\zeta^Y_j}$ for $j\geq 1$ are regeneration times for X; $\varrho_j\,{:\!=}\,X_{\zeta^X_j}=Y_{\zeta^Y_j}$ are regeneration points. An important result that we will utilise throughout this section is that the time and distance between regenerations of Y have exponential moments. That is, there exists $c>0$ such that, for any $j\geq 1$,

(3.1) \begin{equation}{\textit{E}}[e^{c(\varrho_j-\varrho_{j-1}\!)}]\leq {\textit{E}}\Big[e^{c(\!\zeta^Y_j-\zeta^Y_{j-1}\!)}\Big]<\infty,\end{equation}

which is a direct consequence of [Reference Dembo, Peres and Zeitouni20, Lemma 5.1].

Write

\begin{align*}\chi^+_j & \;\,{:\!=}\; \nu_\beta\left(\Big(\zeta_j^X-\zeta_{j-1}^X\Big)-\overline{\mu}\Big(\zeta^Y_j-\zeta^Y_{j-1}\Big)\right)\!, \\ \chi^-_j & \;\,{:\!=}\; \upsilon_\beta\Big(\zeta^Y_j-\zeta^Y_{j-1}\Big)-(\varrho_j-\varrho_{j-1}), \\ \chi_j&\;\,{:\!=}\; \nu_\beta\left(\zeta^X_j-\zeta_{j-1}^X\right)-(\varrho_j-\varrho_{j-1})\;=\;\chi^+_j+\chi^-_j.\end{align*}

Similarly to [Reference Bowditch17, Lemma 2.3] and using a law of large numbers, we have that the families $\{\chi_j^+\}_{j\geq 2}$, $\{\chi_j^-\}_{j\geq 2}$, and $\{\chi_j\}_{j\geq 2}$ are centred and i.i.d. under ${\mathbb{P}}$ whenever $\beta>1$ and $\overline{\mu}<\infty$. In particular,

\begin{equation*}\Sigma_{m}\,{:\!=}\, \sum_{j=2}^{m}\chi_j=\left(\zeta_{m}^X\nu_\beta-X_{\zeta_{m}^X}\right) -\left(\zeta_1^X\nu_\beta-X_{\zeta_1^X}\right)\end{equation*}

is a sum of i.i.d. centred random variables, which we will later use to approximate $\Delta_{nt\nu_\beta}-nt$ along a certain time change. We first show that $\Sigma_{n_kt}/a_{n_k}$ converges to a Lévy process. We begin, in Lemma 3.1, by showing that $\{\chi_j^-\}_{j\geq 2}$ are negligible.

Lemma 3.1. Suppose $\beta>1$ and $T<\infty$. If $\lim_{n\rightarrow\infty}n^{-1/2}a_n=\infty$ then

\begin{equation*}\sup_{t\leq T}\;\left|\sum_{j=2}^{\lfloor nt\rfloor}\frac{\chi_j^-}{a_n}\right|\end{equation*}

converges in ${\mathbb{P}}$-probability to 0 as ${n \rightarrow \infty}$.

Proof. Define

\begin{equation*} \Sigma_m^-\,{:\!=}\sum_{j=2}^m\chi_j^-, \end{equation*}

which is a martingale with centred and independent jumps which, by (3.1), have finite exponential moments.

By Kolmogorov’s inequality we then have that for $\varepsilon>0$,

\begin{equation*}{\mathbb{P}}\left(\sup_{m\leq nT}|\Sigma_m^-|\geq \varepsilon a_n\right) \;\leq\; \frac{{\mathbb{E}}[(\Sigma_{nT}^-)^2]}{\varepsilon^2a_n^2} \;\leq\; \frac{C_{T,\varepsilon}n}{a_n^2}{\mathbb{E}}\left[\left(\chi_2^-\right)^2\right]\!,\end{equation*}

which converges to 0 as ${n \rightarrow \infty}$ since $n/a_n^2\rightarrow 0$ and $\chi_2^-$ has exponential moments.□

We now turn to proving convergence of $\Sigma_{m}^+\,{:\!=}\sum_{j=2}^{m}\chi_j^+$ after suitable scaling, which we do by considering the Laplace transform in Proposition 3.1. We first prove a technical lemma that will allow us to use dominated convergence in the proof of this proposition.

Lemma 3.2. Under the assumptions of Theorem 1.2 there exist constants c, C depending only on $\lambda$ such that for sufficiently large k (independently of $\ell$),

\begin{equation*}c\ell \; \leq \; n_k\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_{n_k}}(\eta_0-\overline{\mu})\right)\right]^\ell\right]\right) \; \leq \; C\ell^2.\end{equation*}

Proof. For simplicity we write n for $n_k$, with the understanding that we consider subsequences along which (1.4) holds. By Jensen’s inequality, for a positive random variable Z and $\ell \in {\mathbb{Z}}$ satisfying ${\mathbb{E}}[Z^\ell]<\infty$ we have

(3.2) \begin{equation}\ell\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[Z\right]\right]\right)\;\leq \; \log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[Z\right]^\ell\right]\right) \; \leq \; \log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[Z^\ell\right]\right]\right)\!.\end{equation}

Choose

\begin{equation*}Z(\lambda)=\exp\left(-\frac{\lambda}{a_n}(\eta_0-\overline{\mu})\right)\!; \end{equation*}

then

\[Z(\lambda)^\ell=\exp\left(-\frac{\ell\lambda}{a_n}(\eta_0-\overline{\mu})\right)\!.\]

By (1.4) and the lower bound in (3.2) we then have that

\begin{equation*}c\ell\lambda^\alpha\leq\ell f(1,\lambda)\leq f(\ell,\lambda)\end{equation*}

for a positive constant c. In particular, $f(\ell,\lambda)>0$ for all $\lambda>0$. By (3.2),

\begin{equation*}n\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_n}(\eta_0-\overline{\mu})\right)\right]^\ell\right]\right) \geq\ell n\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_n}(\eta_0-\overline{\mu})\right)\right]\right]\right)\!,\end{equation*}

which is bounded below by $c_\lambda \ell$ using (1.4) with $\ell=1$. This completes the lower bound.

For the upper bound, using (3.2) we have that

(3.3) \begin{equation}n\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_n}(\eta_0-\overline{\mu})\right)\right]^\ell\right]\right) \leq \frac{n\lambda \ell\overline{\mu}}{a_n}+n\log\left({\mathbb{E}}\left[\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)\right]\right)\!. \end{equation}

Using that $e^{-y}-1+y\leq y\wedge y^2/2$ for $y\geq 0$, we have

\begin{align*} &{\mathbb{E}}\left[\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)\right] - 1+\frac{\lambda \ell}{a_n}\overline{\mu} \\ &\quad = {\mathbb{E}}\left[\left(\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)-1+\frac{\lambda \ell}{a_n}\overline{\mu}\right){\textbf{1}}_{\{\eta_0>a_n\}}\right]\\ &\qquad+{\mathbb{E}}\left[\left(\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)-1+\frac{\lambda \ell}{a_n}\overline{\mu}\right){\textbf{1}}_{\{\eta_0\leq a_n\}}\right] \\ &\quad \leq \lambda \ell{\mathbb{E}}\left[\frac{\eta_0}{a_n}{\textbf{1}}_{\{\eta_0> a_n\}}\right]+\left(\lambda \ell\right)^2{\mathbb{E}}\left[\left(\frac{\eta_0}{a_n}\right)^2{\textbf{1}}_{\{\eta_0\leq a_n\}}\right]\!.\end{align*}

By (1.4) with $\ell=1$ we have that, under the annealed law, $\eta_0$ belongs to the domain of partial attraction of an infinitely divisible law with scaling sequence $a_n$ (see [Reference Gnedenko and Kolmogorov29, Chapter 7]), so

\begin{equation*}n{\mathbb{E}}\left[ \frac{\eta_0}{a_n}{\textbf{1}}_{\{\eta_0\geq a_n\}}\right]\quad \text{and}\quad n{\mathbb{E}}\left[ \left(\frac{\eta_0}{a_n}\right)^2{\textbf{1}}_{\{\eta_0\leq a_n\}}\right]\end{equation*}

are bounded above. We therefore have that, for some constant C,

(3.4) \begin{equation}{\mathbb{E}}\left[\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)\right] \leq 1 - \frac{\lambda \ell \overline{\mu}}{a_n} +\frac{C\ell^2}{n}\end{equation}

uniformly over $\ell\geq 1$.

We want to show that we can choose $N_0$ sufficiently large so that the right-hand side of (3.4) is strictly larger than zero for any $k\geq 1$ and $n\geq N_0$. Since $\alpha\in(1,2)$ we have that $n/a_n^2\rightarrow 0$ as ${n \rightarrow \infty}$; therefore fix $N_0$ large enough that, for $n\geq N_0$,

\begin{equation*}\frac{n(\lambda\overline{\mu})^2}{Ca_n^2}<\frac{1}{2}.\end{equation*}

We can write

\begin{equation*}1 - \frac{\lambda \ell \overline{\mu}}{a_n} +\frac{C\ell^2}{n}=1-\frac{\ell}{a_n}\left(\lambda\overline{\mu}-\frac{C\ell a_n}{n}\right)\!,\end{equation*}

where the term in the brackets is only positive if $\ell<\lambda\overline{\mu}n/(Ca_n)$. This means that

\begin{equation*}\frac{\ell}{a_n}\left(\lambda\overline{\mu}-\frac{Cka_n}{n}\right) \;\leq\; \frac{\lambda\overline{\mu}n}{Ca_n^2}\cdot\lambda\overline{\mu} \;\leq\; \frac{1}{2}\end{equation*}

for any $n\geq N_0$. This shows that the right-hand side of (3.4) is strictly larger than zero for any $\ell\geq 1$ and $n\geq N_0$.

Since $\log(1+x)\leq x$ for $x>-1$, we then have that

\begin{equation*}n\log\left({\mathbb{E}}\left[\exp\left(-\frac{\lambda \ell}{a_n}\eta_0\right)\right]\right)\leq - \frac{n\lambda \ell\overline{\mu}}{a_n} +C\ell^2.\end{equation*}

Combining this with (3.3), we have that

\begin{align*}n\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_n}(\eta_0-\overline{\mu})\right)\right]^\ell\right]\right)\leq C\ell^2.\\[-42pt]\end{align*}

□

We now show that the scaled sum of the random variables $\chi_j^+$ converges along the given subsequences. Let

\[{\mathcal{I}}\,{:\!=}\{Y_j\}_{j=\zeta^Y_1}^{\zeta^Y_2-1}\]

denote the set of vertices in the first regeneration block, and for $\ell=1,2,\ldots$ let ${\mathcal{I}}_\ell\,{:\!=}\{x \in {\mathcal{I}}\;: \; {\mathcal{L}}(x,\infty)=\ell\}$ denote those visited exactly $\ell$ times.

Proposition 3.1. Under the assumptions of Theorem 1.2,

\begin{equation*} \lim_{k\rightarrow \infty}{\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_{n_k}}\sum_{j=2}^{\lfloor tn_k\rfloor}\chi^+_j\right)\right]=\exp\left(t\sum_{\ell=1}^\infty f(\ell,\lambda\nu_\beta){\textit{E}}[|{\mathcal{I}}_\ell|]\right)\!. \end{equation*}

Proof. As in Lemma 3.2 we write n for $n_k$, with the understanding that we consider subsequences along which (1.4) holds. Since $\chi_j^+$ are i.i.d. we have that

\begin{equation*}{\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_n}\sum_{j=2}^{\lfloor tn\rfloor}\chi^+_j\right)\right] = \left(\frac{n{\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_n}\chi^+_2\right)-1\right]}{n}+1\right)^{\lfloor tn\rfloor-1};\end{equation*}

therefore it suffices to show instead that

\begin{equation*}{\lim_{{n \rightarrow \infty}}} n{\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_n}\chi^+_2\right)-1\right]=\sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}[|{\mathcal{I}}_\ell|].\end{equation*}

We first show that ${\textit{E}}\left[|{\mathcal{I}}_\ell|\right]$ decays exponentially in $\ell$. Since $\varrho_1\geq 1$ we have that

\begin{equation*}{\textit{E}}\left[|{\mathcal{I}}_\ell|\right] \; \leq \; {\textit{E}}\left[\sum_{x \in {\mathcal{I}}}{\textbf{1}}_{\{{\mathcal{L}}(x,\infty)\geq\ell\}}\right] \; = \; \sum_{x \in {\mathbb{N}}}{\textit{E}}\left[{\textbf{1}}_{\{x \in {\mathcal{I}}\}}{\textbf{1}}_{\{{\mathcal{L}}(x,\infty)\geq\ell\}}\right]\!.\end{equation*}

Using Cauchy–Schwarz and the fact that the ${\mathcal{L}}(x,\infty)$ are identically distributed for $x \in {\mathbb{N}}$, we have that this is bounded above by

\begin{equation*}\sum_{x \in {\mathbb{N}}}{\textit{P}}\left(x \in {\mathcal{I}}\right)^{1/2}{\textit{P}}\left({\mathcal{L}}(x,\infty)\geq\ell\right)^{1/2} \; = \; {\textit{P}}\left({\mathcal{L}}(0,\infty)\geq\ell\right)^{1/2} \sum_{x \in {\mathbb{N}}}{\textit{P}}\left(x \in {\mathcal{I}}\right)^{1/2}.\end{equation*}

The number of returns to the origin is geometrically distributed with probability $\upsilon_\beta$ of moving away from the origin and never returning; therefore, ${\textit{P}}\left({\mathcal{L}}(0,\infty)\geq\ell\right)= (1-\upsilon_\beta)^{\ell-1}$. By (3.1) we have that ${\textit{P}}\left(x \in {\mathcal{I}}\right)\leq {\textit{P}}(x \leq \zeta^Y_2) \leq Ce^{-cx}$. It follows that

(3.5) \begin{equation}{\textit{E}}\left[|{\mathcal{I}}_\ell|\right] \; \leq \; C\left(1-\upsilon_\beta\right)^{\ell/2}\sum_{x\geq 0}e^{-cx/2} \; \leq \; C\left(\frac{2}{\beta+1}\right)^{\ell/2}.\end{equation}

Conditional on Y, the holding times at different vertices are independent; therefore

\begin{align*}{\mathbb{E}}\left[\exp\left(-\frac{\lambda }{\nu_\beta a_n}\chi_2^+\right)\right]&={\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_n}\Big(\zeta_2^X-\zeta_1^X-{\mathbb{E}}[\eta_0]\Big(\zeta^Y_2-\zeta^Y_1\Big)\Big)\right)\right] \\ &={\mathbb{E}}\left[\exp\left(-\frac{\lambda}{a_n}\left(\sum_{x\in{\mathcal{I}}}\sum_{j=1}^{{\mathcal{L}}(x,\infty)}(\eta_{x,j}-{\mathbb{E}}[\eta_{x,j}])\right)\right)\right]\\&={\textit{E}}\left[\prod_{x\in{\mathcal{I}}}{\mathbb{E}}^Y\left[\exp\left(-\frac{\lambda}{a_n}\left(\sum_{j=1}^{{\mathcal{L}}(x,\infty)}(\eta_{x,j}-{\mathbb{E}}[\eta_{x,j}])\right)\right)\right]\right]\!. \end{align*}

Under ${\textit{P}}^{\omega,Y}$ the holding times at a given vertex are independent and identically distributed; therefore we can write the above expression as

\begin{equation*} {\textit{E}}\!\left[\prod_{x \in {\mathcal{I}}} {\textbf{E}}\left[ {\textit{E}}^{\omega,Y}\left[\prod_{j=1}^{{\mathcal{L}}(x,\infty)} e^{-\frac{\lambda}{a_n}\left(\eta_{x,j}-{\mathbb{E}}[\eta_{x,j}]\right)}\right]\right]\right] = {\textit{E}}\!\left[\prod_{k=1}^\infty\prod_{x \in {\mathcal{I}}_\ell} {\textbf{E}}\left[ {\textit{E}}^{\omega,Y}\left[e^{-\frac{\lambda}{a_n}\left(\eta_{x,1}-{\mathbb{E}}[\eta_{x,1}]\right)}\right]^\ell\right]\right]\!. \end{equation*}

For $x \in {\mathbb{Z}}$ let $\tilde{\eta}_x=\eta_{x,1}$ be the first holding time at x. Notice that, since $\tilde{\eta}_x$ is independent of Y, we have that

\begin{equation*}\Psi_{\ell,x}\,{:\!=}\,{\textbf{E}}\left[ {\textit{E}}^{\omega,Y}\left[e^{-\frac{\lambda}{a_n}\left(\eta_{x,1}-{\mathbb{E}}[\eta_{x,1}]\right)}\right]^\ell\right]={\textbf{E}}\left[ {\textit{E}}^{\omega}\left[e^{-\frac{\lambda}{a_n}\left(\tilde{\eta}_x-{\mathbb{E}}[\tilde{\eta}_x]\right)}\right]^\ell\right]\end{equation*}

is independent of Y and, therefore, deterministic. In particular, since the $\omega_x$ are i.i.d. under ${\mathbb{P}}$, we have that $\Psi_{\ell,x}$ does not depend on x, and we can write

(3.6) \begin{equation} {\mathbb{E}}\left[\exp\left(-\frac{\lambda}{ \nu_\beta a_n}\chi_2^+\right)\right] = {\textit{E}}\left[\exp\left(\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right)\right]\!.\end{equation}

Furthermore, since $\tilde{\eta}_0\geq0$ we have that $\log(\Psi_{\ell,0}) \leq\ell\lambda{\mathbb{E}}[\eta_0]/a_n$.

Using a Taylor expansion we have that for any $x \in {\mathbb{R}}_+$ there exists $w\in[0,x]$ such that

\begin{equation*}e^x=1+x+\frac{e^w}{2}x^2.\end{equation*}

Note that $\log(\Psi_{\ell,0}) \leq\ell\lambda\overline{\mu}/a_n$ and, by Lemma 3.2, $\log(\Psi_{\ell,0})\geq 0$ for n sufficiently large (independently of $\ell$). It follows from this that there exists a positive random variable W, which is bounded above by

\begin{equation*}\frac{\lambda\overline{\mu}}{a_n}\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|,\end{equation*}

such that

\begin{equation*} n{\textit{E}}\left[\exp\left(\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right)-1\right] = n{\textit{E}}\left[\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})+\frac{e^W}{2}\left(\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right)^2\right]\!.\end{equation*}

Using the previous upper bound on $\log(\Psi_{\ell,0})$ and Cauchy–Schwarz, we then have that

\begin{align*}n{\textit{E}}\!\left[\frac{e^W}{2}\left(\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right)^2\right]& \leq \frac{C_\lambda n}{a_n^2}{\textit{E}}\!\left[\exp\left(\frac{\lambda\overline{\mu}}{a_n}\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|\right)\left(\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|\right)^2\right] \\ & \leq \frac{C_\lambda n}{a_n^2}{\textit{E}}\!\left[\exp\left(\frac{2\lambda\overline{\mu}}{a_n}\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|\right)\right]^{1/2}{\textit{E}}\!\left[\left(\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|\right)^4\right]^{1/2}.\end{align*}

The sum $\sum_{\ell=1}^\infty \ell|{\mathcal{I}}_\ell|$ is the time between regenerations of the embedded walk Y. By (3.1) this has exponential moments, so the second expectation in the previous equation is finite and we can choose n sufficiently large so that the first expectation is also finite. Since $a_n=n^{1/\alpha}L(n)$ for $\alpha \in (1,2)$, we have that $n/a_n^2$ converges to 0 as ${n \rightarrow \infty}$, and therefore

\begin{equation*} n{\textit{E}}\left[\exp\left(\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right)-1\right]= n{\textit{E}}\left[\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right]+o(1).\end{equation*}

The function $n\log(\Psi_{\ell,0})$ is deterministic and converges to the positive constant $f(\ell,\lambda)$. By Lemma 3.2 we have that, for n suitably large, $c\ell\leq n\log(\Psi_{\ell,0})\leq C\ell^2$ independently of $\ell$. It follows from (3.5) that

\begin{equation*}\sum_{\ell=1}^\infty \ell^2{\textit{E}}\left[|{\mathcal{I}}_\ell|\right]<\infty,\end{equation*}

and therefore by dominated convergence we have that, as ${n \rightarrow \infty}$,

\begin{equation*} n{\textit{E}}\left[\sum_{\ell=1}^\infty |{\mathcal{I}}_\ell|\log(\Psi_{\ell,0})\right] \; = \; \sum_{\ell=1}^\infty {\textit{E}}\left[|{\mathcal{I}}_\ell|\right] n\log(\Psi_{\ell,0}) \; \rightarrow \; \sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}\left[|{\mathcal{I}}_\ell|\right]\!.\end{equation*}

Combining with (3.6) completes the proof.□

By Lemma 3.1 and Proposition 3.1 we now have the following corollary.

Corollary 3.1. Under the assumptions of Theorem 1.2, the sequence $\Sigma_{n_kt}/a_{n_k}$ converges in distribution on $D_{J_1}$ to a process $\tilde{{\mathcal{V}}}_t$ with Laplace exponent

\[t\sum_{\ell=1}^\infty f(\ell,\lambda\nu_\beta){\textit{E}}[|{\mathcal{I}}_\ell|].\]

We now have the required framework to prove Theorem 1.2, which is a convergence result for

\[(\Delta_{n_kt\nu_\beta}-n_kt)/a_{n_k}\]

and

\[(X_{n_kt}-n_kt\nu_\beta)/a_{n_k}.\]

The approach we take is to compare $\Delta$ with $\Sigma$ and then use a result of [Reference Whitt44] that allows us to deduce convergence for the inverse of $\Delta$.

Proof of Theorem 1.2. Write $\psi_t\,{:\!=}\sup\{j\geq 0\;:\;\zeta^X_j\leq t\}$ to denote the number of regenerations by time t. It follows by the law of large numbers and continuity of the inverse at strictly increasing functions [Reference Whitt44, Corollary 13.6.4] that $\psi_{nt}n^{-1}$ converges ${\mathbb{P}}$-a.s. to $R_t\,{:\!=}\,(\overline{\mu}{\mathbb{E}}[\zeta^Y_2-\zeta^Y_1])^{-1}t$. By Corollary 3.1 and $J_1$-continuity of composition [Reference Whitt44, Theorem 13.2.2] we then have that

\[\Sigma_{\psi_{tn_k}}/a_{n_k}\]

converges in distribution on $D_{J_1}([0,\infty),{\mathbb{R}})$ to the process $\tilde{{\mathcal{V}}}_{R_t}$.

Recall that

\begin{equation*}\Sigma_{m}=\left(\zeta^X_{m}\nu_\beta-X_{\zeta^X_{m}}\right) -\left(\zeta_1^X\nu_\beta-\varrho_1\right)\!,\end{equation*}

where $a_{n_k}^{-1}\left(\zeta_1^X\nu_\beta-\varrho_1\right)$ converges to 0 in probability. It follows that

\begin{equation*}\Sigma_{\psi_{n_kt}}+ \left(\zeta_1^X\nu_\beta-\varrho_1\right) \; = \; \nu_\beta\zeta^X_{\psi_{n_kt}}-X_{\zeta^X_{\psi_{n_kt}}} \; = \; \nu_\beta\left(\Delta_{\varrho_{\psi_{n_kt}}}-\varrho_{\psi_{n_kt}}\nu_\beta^{-1}\right)\!.\end{equation*}

Notice that

\[\varrho_{\psi_{n_kt}}n_k^{-1}\]

is bounded above by t and converges ${\mathbb{P}}$-a.s. to t, since the distance between regenerations has exponential moments by (3.1). In particular, if

\begin{equation*}u_k\,{:\!=}\,d_{M_1}\left(\left(\frac{\Delta_{\nu_\beta\varrho_{\psi_{n_kt}}}-\varrho_{\psi_{n_kt}}}{a_{n_k}}\right)_{t\in[0,T]},\left(\frac{\Delta_{n_kt\nu_\beta}-n_kt}{a_{n_k}}\right)_{t\in[0,T]}\right)\end{equation*}

converges to 0 in probability, then $(\Delta_{n_kt\nu_\beta}-n_kt)/a_{n_k}$ converges in distribution on $D_{M_1}$ to the process

\[{\mathcal{V}}_t\,{:\!=}\nu_\beta^{-1}\tilde{{\mathcal{V}}}_{R_{\nu_\beta t}}.\]

Since $\Delta$ is increasing and the regeneration points are ordered, we have that

\begin{equation*}\varrho_{\psi_{n_kt}}\leq n_kt \leq \varrho_{\psi_{n_kt}+1} \qquad \text{ and } \qquad \Delta_{\varrho_{\psi_{n_kt}}}\leq \Delta_{n_kt} \leq \Delta_{\varrho_{\psi_{n_kt}+1}}.\end{equation*}

In particular,

\[u_k\leq \frac{(1+\nu_\beta)(\varrho_{\psi_{n_kt}+1}-\varrho_{\psi_{n_kt}})}{a_{n_k}}.\]

Using that there are at most $n_kT$ regenerations by level $n_kT$, we then have that $u_k$ is bounded above by $C\sup_{j\leq n_kT}(\varrho_{j+1}-\varrho_j)/a_{n_k}.$ Let $\varepsilon>0$; using a union bound and that the distance between regenerations have exponential moments, we have that

\begin{equation*} {\mathbb{P}}\left(u_k >\varepsilon\right) \;\leq \; n_kT{\textit{P}}(\varrho_2-\varrho_1>C\varepsilon a_{n_k}) +{\textit{P}}(\varrho_1>C\varepsilon a_{n_k}) \; \leq \; C_Tn_ke^{-c_\varepsilon a_{n_k}},\end{equation*}

which converges to 0 as $k\rightarrow \infty$.

The inverse $\Delta_{t}^{-1}=\sup\{X_s: \; s\leq t\}=:\overline{X}_t$ is the farthest vertex reached by time t. By inverse with linear centring [Reference Whitt44, Theorem 13.7.1], we have that

\begin{equation*}\left(\frac{\Delta_{n_kt\nu_\beta}-n_kt}{a_{n_k}}, \frac{\overline{X}_{n_kt}-n_kt\nu_\beta}{a_{n_k}}\right)_{t\geq 0}\end{equation*}

converge jointly in distribution on $D_{M_1}\times D_{M_1}$ to the pair $({\mathcal{V}}_t,-\nu_{\beta}{\mathcal{V}}_{t})$. Moreover, letting $\tau_0$ denote the first hitting time of 0, using a union bound we have

\begin{equation*} {\mathbb{P}}\left(\sup_{t\leq nT}(\overline{X}_t-X_t)>C\log(n)\right) \;\leq\; nT{\textit{P}}_{\lfloor C\log(n)\rfloor}(\tau_0<\infty) \;=\; nT\beta^{-\lfloor C\log(n)\rfloor},\end{equation*}

which converges to 0 for C large. Therefore we have the desired convergence replacing $\overline{X}$ with X.

Finally, as in the proof of Theorem 1.1, independent and stationary increments of ${\mathcal{V}}_t$ follow from independent and stationary increments of the discrete process. Combined with the form of the Laplace transform we have that ${\mathcal{V}}_t$ is a Lévy process. Similarly, when the convergence holds along all subsequences $n_k$, the form of the Laplace transform yields self-similarity and therefore the process is $\alpha$-stable.□

Noting that

\[{\mathcal{V}}_t\,{:\!=}\nu_\beta^{-1}\tilde{{\mathcal{V}}}_{R_{\nu_{\beta}t}},\]

where $R_t=(\overline{\mu}{\mathbb{E}}[\zeta^Y_2-\zeta^Y_1])^{-1}t$ and $\tilde{{\mathcal{V}}}_{t}$ has the Laplace exponent given in Corollary 3.1, we can express the Laplace transform of ${\mathcal{V}}_t$ as

(3.7) \begin{equation}{\mathbb{E}}[e^{-\lambda{\mathcal{V}}_t}]\;=\;\exp\left(\frac{t}{\overline{\mu}{\mathbb{E}}[\zeta^Y_2-\zeta^Y_1]}\sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}[|{\mathcal{I}}_\ell|]\right)\!.\end{equation}

In particular, note that

\begin{equation*}\sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}[|{\mathcal{I}}_\ell|]\asymp \lambda^\alpha\end{equation*}

as $\lambda\rightarrow 0$. We therefore have that

\begin{equation*}\lim_{\lambda\rightarrow 0}\lambda^{-1}\sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}[|{\mathcal{I}}_\ell|]=0 \quad \text{and} \quad \lim_{\lambda\rightarrow 0}\lambda^{-2}\sum_{\ell=1}^\infty f(\ell,\lambda){\textit{E}}[|{\mathcal{I}}_\ell|]=\infty,\end{equation*}

which shows that ${\mathcal{V}}_t$ is centred but heavy-tailed (in particular, not a Brownian motion). Furthermore, since the Laplace transform exists for all $\lambda>0$, we have that ${\mathcal{V}}_t$ has a light negative tail and therefore must have a heavy positive tail. This coincides with what we would expect from the discrete process, which was a one-sided jump process with drift.

We now prove a technical result that will make the condition (1.4) more straightforward to apply. We show that we can decompose the excursion times and remove parts of the excursion that do not contribute to the fluctuations. Let $\tilde{X}$ be an RTRW with some trap measure $\tilde{\pi}$ and the same embedded walk Y that is associated to X, then denote by $\tilde{\eta}$ the sequence of holding times for $\tilde{X}$.

Corollary 3.2. Let $\beta>1$, $\alpha\in(1,2)$, and $f:{\mathbb{N}}\times{\mathbb{R}}_+\rightarrow {\mathbb{R}}$ be such that $f(1,\lambda)\asymp\lambda^\alpha$. Suppose that the exists a coupling of $\eta$, $\tilde{\eta}$ such that, for some $\epsilon>0$, ${\mathbb{E}}[|\eta_{0,1}-\tilde{\eta}_{0,1}|^{\alpha+\epsilon}]<\infty$. If there exists $n_k\nearrow \infty$, $a_n=n^{1/\alpha}L(n)$ for some L slowly varying such that for any $\ell \in{\mathbb{N}}$ and $\lambda>0$ we have

\begin{equation*} n_k\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{a_{n_k}}(\tilde{\eta}_0-{\mathbb{E}}[\tilde{\eta}_0])\right)\right]^\ell\right]\right)\sim\, f(\ell,\lambda) \end{equation*}

as $k\rightarrow \infty$, then, as $k\rightarrow \infty$,

\begin{equation*}\left(\frac{\Delta_{n_kt\nu_\beta}-n_kt}{a_{n_k}}, \frac{X_{n_kt}-n_kt\nu_\beta}{a_{n_k}}\right)_{t\geq 0}\end{equation*}

converges in ${\mathbb{P}}$-distribution on $D_{M_1}\times D_{M_1}$ to $({\mathcal{V}}_t, -\nu_{\beta}{\mathcal{V}}_t)_{t\geq 0}$, where ${\mathcal{V}}_t$ is a Lévy process. In particular, if (1.4) holds for all subsequences $n_k$, then $f(\ell,\lambda)=g(\ell)\lambda^\alpha$ for some $g\not\equiv0$, and ${\mathcal{V}}_t$ is an $\alpha$-stable process.

Proof. Since, in each model, the traps are i.i.d. under ${\mathbb{P}}$ and each holding time in a fixed trap is independent, by translation-invariance there exists a version $\tilde{X}$ coupled to X such that ${\mathbb{E}}[(\eta_k-\tilde{\eta}_k)^{\alpha+\epsilon}]<\infty$ for all $k \in {\mathbb{N}}$. Let $\tilde{\Delta}_m\,{:\!=}\inf\{t>0:\tilde{X}_t=m\}$ denote the first hitting time of level m by the walk $\tilde{X}$ and $\tilde{\nu}_\beta$ its speed. We want to show that

(3.8) \begin{equation}\sup_{m\leq n}\left|\frac{\tilde{\Delta}_m-\Delta_m-m\big(\tilde{\nu}_\beta^{-1}-\nu_\beta^{-1}\big)}{a_n}\right|\end{equation}

converges to 0 in probability as ${n \rightarrow \infty}$.

First suppose that $\tilde{\eta}_k\geq \eta_k$ ${\mathbb{P}}$-a.s. In particular, we assume that ${\mathbb{E}}[\tilde{\eta}_0-\eta_0]>0$, since otherwise the result follows trivially. Define $\hat{X}$ to be the RTRW with embedded walk Y and holding times $\tilde{\eta}_k-\eta_k$. This walk has first hitting times $\hat{\Delta}_m=\tilde{\Delta}_m-\Delta_m$, and since ${\mathbb{E}}[\hat{\eta}_0^{\alpha+\epsilon}]<\infty$, we have that the walk is ballistic and has speed $\hat{\nu}_\beta=\upsilon_\infty/{\mathbb{E}}[\tilde{\eta}_0-\eta_0]$ (see (1.2)), which gives us that $\hat{\nu}_\beta^{-1}=\tilde{\nu}_\beta^{-1}-\nu_\beta^{-1}$. In particular, (3.8) is equal to

\begin{equation*}\sup_{m\leq n}\left|\frac{\hat{\Delta}_m-m\hat{\nu}_\beta^{-1}}{a_n}\right|.\end{equation*}

It then follows straightforwardly from the proof of Theorem 1.2 that this converges to 0 in probability as ${n \rightarrow \infty}$.

Note that by a symmetric argument we have that if $\tilde{\eta}_k\leq \eta_k$ ${\mathbb{P}}$-a.s. then (3.8) converges to 0 in probability as ${n \rightarrow \infty}$. For general $\tilde{X}$ we can define a new RTRW with holding times $\tilde{\eta}_k\land \eta_k$. Using the triangle inequality we then have that (3.8) converges to 0 in probability as ${n \rightarrow \infty}$. The result then follows by Theorem 1.2.□

4. Counterexample for convergence in the Skorokhod topology

Recall that in Theorem 1.2 we consider the $M_1$ topology as opposed to the stronger $J_1$ topology considered for $\Sigma$ in Corollary 3.1. Given that our approach was to compare $\Delta$ with $\Sigma$, one may question whether we should be able to extend the statements of Theorem 1.2 to the $J_1$ topology. This is not possible in the general setting because, between regenerations, the walk experiences several large holding times at the same significant vertex. This means that $\Delta$ fluctuates between regeneration points, whereas $\Sigma$ groups the fluctuations into a single jump.

We now use the example of the random walk in random scenery to show that, in general, the convergence in Theorem 1.2 cannot be extended to $J_1$ convergence. Let $(\kappa_x)_{x\in {\mathbb{Z}}}$ be an i.i.d. sequence of $(0,\infty)$-valued random variables under ${\mathbb{P}}$. We then define the holding times by ${\textit{P}}^\omega(\eta_{x,j}=\kappa_x)=1$ for all $j\geq 1$ and $x \in {\mathbb{Z}}$. Let X denote the RTRW, S its clock process, and $\Delta$ the first hitting times.

Proposition 4.1 shows that the assumptions of Theorems 1.1 and 1.2 hold under a simple stable law condition for the environment.

Proof. The holding time $\eta_0$ is a.s. fixed and equal to $\kappa_0$ under ${\textit{P}}^\omega$; we therefore have that ${\textit{E}}^\omega[g(\eta_0)]=g(\kappa_0)$ for any function g.

For $\alpha \in (0,1)$, by assumption, $\kappa_0$ belongs to the domain of attraction of a stable law of index $\alpha$ with respect to ${\mathbb{P}}$; hence there exists $a_n=n^{1/\alpha}L(n)$ such that $n{\mathbb{P}}(\kappa_0>ta_n)\sim Ct^{-\alpha}$. Since $\kappa_0$ is a.s. non-negative, it follows from a Tauberian theorem (e.g. [Reference Feller22, Theorem XIII.5.1]) that

\begin{equation*} -n\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right) \;=\; -n\log\left({\textbf{E}}\left[\exp\left(-\frac{\ell\lambda}{a_n}\kappa_0\right)\right]\right)\sim\, C(\ell\lambda)^\alpha.\end{equation*}

The first result then follows from Theorem 1.1.

Similarly, for $\alpha \in (1,2)$, by assumption, $\kappa_0$ belongs to the domain of attraction of a stable law of index $\alpha$ with respect to ${\mathbb{P}}$; hence there exists $a_n=n^{1/\alpha}L(n)$ such that $n{\mathbb{P}}(\kappa_0-{\textbf{E}}[\kappa_0]>ta_n)\sim Ct^{-\alpha}$, and therefore, since $\kappa_0-{\textbf{E}}[\kappa_0]$ is centred and bounded below,

\begin{equation*}n\log\left({\textbf{E}}\left[\varphi_{\omega_0}\left(\frac{\lambda}{a_n}\right)^\ell\right]\right)+n\ell\overline{\mu}/a_n = n\log\left({\textbf{E}}\left[\exp\left(-\frac{\ell\lambda}{a_n}(\kappa_0-{\textbf{E}}[\kappa_0])\right)\right]\right)\end{equation*}

converges to $C(\ell\lambda)^\alpha$ for some constant C. The result then follows from Theorem 1.2.□

We now give an argument which shows that, in general, the $M_1$ convergence in Theorem 1.2 cannot be extended to $J_1$ convergence. For $f \in D([0,\infty),{\mathbb{R}})$ and $T,h>0$, let

\begin{equation*}\tilde{\omega}(\,f,T,h)\,{:\!=}\inf_{(I_k)}\max_k\sup_{r,s \in I_k}|f(s)-f(r)|\end{equation*}

denote the modulus of continuity, where the infimum is over partitions of [0, T) into $I_k=[v_k,v_{k+1})$ such that $v_{k+1}-v_k>h$ for all $k\geq 1$. By [Reference Kallenberg32, Theorem 16.10], if $Z,(Z^n)_{n=1}^\infty$ are random elements of $D([0,\infty),{\mathbb{R}})$, then $Z^n$ converges in distribution to Z in $D_{J_1}$ if and only if $Z^n$ converges to Z in the sense of finite-dimensional distributions and, for any $T>0$,

\begin{equation*}\lim_{h\rightarrow 0}\limsup_{{n \rightarrow \infty}}{\mathbb{E}}[\tilde{\omega}(Z^n,T,h)\land 1]=0.\end{equation*}

Let us assume that $\beta>1$ and ${\mathbb{P}}(\kappa_0\geq t) = t^{-\alpha}\land 1$ for $\alpha \in (1,2)$. Write

\begin{equation*}Z^n(t)\,{:\!=}\frac{X_t-t\nu_\beta}{a_n};\end{equation*}

then

\[|Z^n(s)-Z^n(r)|=\left|\frac{X_s-X_r-\nu_\beta(s-r)}{a_n}\right|.\]

Since $\alpha>1$ we have that the walk is ballistic; therefore,

\begin{equation*}{\lim_{{n \rightarrow \infty}}}{\mathbb{P}}\left(\Delta_{nT\nu_\beta/2}\leq nT\right)=1.\end{equation*}

That is, with high probability the walk visits at least $nT\nu_\beta/2$ vertices by time nT.

Let $\lambda>0$; then the probability that at least one of these first $nT\nu_\beta/2$ vertices has a holding time of at least $\lambda_n\,{:\!=}\lambda a_n=\lambda n^{1/\alpha}$ and no more than $h_n\,{:\!=}\,hn/3$ is at least

\begin{equation*} {\mathbb{P}}\left(\bigcup_{x=1}^{nT\nu_\beta/2} \{\kappa_x\in [\lambda_n,h_n)\} \right) = 1-\left(1-(\lambda_n)^{-\alpha}+(h_n)^{-\alpha}\right)^{nT\nu_\beta/2} \rightarrow 1-e^{-\frac{\lambda^{-\alpha}T\nu_\beta}{2}}>0.\end{equation*}

Let $\skew2\hat{x}\,{:\!=}\inf\{x\geq 0:\kappa_x \in [\lambda_n,h_n)\}$. Notice that the probability that the embedded walk moves from a vertex y to $y-1$ and then back to y starting from the first hitting time of y is

\begin{equation*}{\mathbb{P}}(Y_{\Delta_y^Y+1}=y-1, \; Y_{\Delta_y^Y+2}=y)=\frac{\beta}{(\beta+1)^2}>0,\end{equation*}

independently of the environment. In particular, this holds for $y=\skew2\hat{x}$, and writing

\begin{equation*}{\mathcal{A}}_n \,{:\!=}\,\{\Delta_{nT\nu_\beta/2}\leq nT\}\;\cap\;\{\skew2\hat{x}\leq nT\nu_\beta/2\} \;\cap\;\Big\{Y_{\Delta_{\skew2\hat{x}}^Y+1}=\skew2\hat{x}-1\Big\}\;\cap\;\Big\{Y_{\Delta_{\skew2\hat{x}}^Y+2}=\skew2\hat{x}\Big\},\end{equation*}

we have that for any fixed $h>0$,

\begin{equation*}\limsup_{{n \rightarrow \infty}}{\mathbb{P}}({\mathcal{A}}_n)\;\geq\; \frac{\beta}{(\beta+1)^2}\left(1-e^{-\frac{\lambda^{-\alpha}T\nu_\beta}{2}}\right)\;>\;0.\end{equation*}

Let $\vartheta_n\,{:\!=}\, \Delta_{\skew2\hat{x}}$ be the first hitting time of $\skew2\hat{x}$; then $\vartheta_n^+\,{:\!=}\, \vartheta_n+\kappa_{\skew2\hat{x}}$ is the time that the walk first moves from $\skew2\hat{x}$ to one of its neighbours. Similarly, let $\tilde{\vartheta}_n\,{:\!=}\, \inf\{t>\vartheta_n^+:X_t=\skew2\hat{x}\}$ denote the first return time to $\skew2\hat{x}$ after the first visit, and let $\tilde{\vartheta}_n^+\,{:\!=}\, \tilde{\vartheta}_n+\kappa_{\skew2\hat{x}}$ be the time at which the walk moves away from $\skew2\hat{x}$ for the second time.

Let $(I_k)$ be a partition of [0,T) into intervals $[v_k,v_{k+1})$ satisfying $v_{k+1}-v_k>h$ for all k. On the event ${\mathcal{A}}_n$ we have that for any such partition there exists some k such that either $\vartheta_n,\vartheta_n^+ \in I_k$ or $\tilde{\vartheta}_n,\tilde{\vartheta}_n^+ \in I_k$. We then have that

\begin{equation*}\limsup_{{n \rightarrow \infty}}|Z^n(\vartheta_n)-Z^n(\vartheta_n^+)|\;=\;\limsup_{{n \rightarrow \infty}}\frac{\nu_\beta\kappa_{\skew2\hat{x}}+1}{a_n}\geq \lambda \nu_\beta.\end{equation*}

The same holds for $\tilde{\vartheta}_n,\tilde{\vartheta}_n^+$. In particular, choosing $\lambda \leq \nu_\beta^{-1}$, we have

\begin{equation*}\limsup_{{n \rightarrow \infty}}{\mathbb{E}}[\tilde{\omega}(Z^n,T,h)\land 1]\;\geq\; \lambda \nu_\beta\limsup_{{n \rightarrow \infty}}{\mathbb{P}}({\mathcal{A}}_n)\; \geq\; \frac{\lambda \nu_\beta\beta\left(1-e^{-\frac{\lambda^{-\alpha}T\nu_\beta}{2}}\right)}{(\beta+1)^2}\;>\;0.\end{equation*}

This proves that

\begin{equation*}\lim_{h\rightarrow 0}\limsup_{{n \rightarrow \infty}}{\mathbb{E}}[\tilde{\omega}(Z^n,T,h)\land 1]\neq0\end{equation*}

for the random walk in random scenery, and therefore Theorem 1.2 cannot be extended to the $J_1$ topology.

5. Random walks on Galton–Watson trees

In this section we study biased random walks on a subcritical GW tree conditioned to survive for the case in which the walk is ballistic and does not satisfy a central limit theorem. Specifically, we prove Theorem 1.3, which is a generalised, subsequential, functional central limit theorem.

It has been shown in [Reference Kesten33] that there is a well defined probability measure ${\mathbb{P}}$ over subcritical GW trees conditioned to survive ${\mathcal{T}}$ which arises as the limit as ${n \rightarrow \infty}$ of probability measures over GW trees conditioned to survive up to generation n. We can construct a random tree with the desired law using two types of vertices, which we refer to as special and normal. Recall that the offspring law is given by $p_k\,{:\!=}\,{\mathbb{P}}(\xi=k)$ and $\mu\,{:\!=}\sum_{k\geq 0}kp_k\in(0,1)$. Define $\xi^*$ to be a random variable with the size-biased law given by the probabilities ${\mathbb{P}}(\xi^*=k)=kp_k\mu^{-1}$. The construction is then as follows:

1. 1. Start with a single special vertex.

2. 2. At each generation, let every normal vertex give birth to vertices according to independent copies of the original offspring distribution (all of which are labelled as normal).

3. 3. At each generation, let every special vertex give birth to vertices according to independent copies of the size-biased distribution (one of which is chosen uniformly at random to be special, while the remaining vertices are labelled as normal).

We will denote by $\rho_k$ the special vertex in the kth generation and by ${\mathcal{Y}}\,{:\!=}\,(\rho_0=\rho,\rho_1,\ldots)$ the semi-infinite path formed by the special vertices (which we call the backbone).

For ${\mathcal{T}}$ fixed, we write $\overleftarrow{x}$ to denote the parent of $x \in {\mathcal{T}}$ and c(x) to denote the set of children of x. The $\beta$-biased random walk X is then the Markov chain started from $\rho$ with transition probabilities

\begin{equation*}{\textit{P}}^{{\mathcal{T}}}_\rho(X_{n+1}=y|X_n=x)=\begin{cases} \dfrac{1}{1+\beta |c(x)|} & \text{if } y=\overleftarrow{x}, \\[10pt] \dfrac{\beta}{1+\beta |c(x)|} & \text{if } y \in c(x), \; x \neq \rho, \\[10pt] \dfrac{1}{|c(\rho)|} & \text{if } y \in c(x), \; x =\rho, \\[10pt] 0 & \text{otherwise.} \\ \end{cases}\end{equation*}

It is natural to consider whether the convergence in Theorem 1.3 can be proved more generally (i.e. along all subsequences); however, this is not the case. To see this we first note that the height ${\mathcal{H}}\,{:\!=}\sup\{n\geq 0:Z_n>0\}$ of the subcritical tree is approximately geometrically distributed with parameter $\mu$; specifically, by [Reference Lyons, Pemantle and Peres36, Theorem B], the sequence ${\mathbb{P}}({\mathcal{H}}\geq n)\mu^{-n}$ is decreasing and converges to a positive constant:

(5.1) \begin{equation}c_\mu\,{:\!=}\,{\lim_{{n \rightarrow \infty}}} \frac{{\mathbb{P}}({\mathcal{H}}\geq n)}{\mu^{n}}.\end{equation}

The time spent in a branch largely consists of a geometric number of excursions from the deepest point to itself. Each of these excursions is short, but by the gambler’s ruin, the probability that the walk escapes the branch on a given excursion is of the order $\beta^{-{\mathcal{H}}}$. It follows that the trapping times behave approximately like $\beta^{{\mathcal{H}}}$. These times do not belong to the domain of attraction of any stable law (as seen in Example 2.1) and therefore we do not observe full convergence. We direct the reader to [Reference Ben Arous, Fribergh, Gantert and Hammond7] for a more detailed illustration of this lattice effect. A related model has been considered in [Reference Ben Arous and Hammond8, Reference Hammond31] in which the bias is chosen randomly for each edge according to a non-lattice condition. This modification removes the lattice effect in the sub-ballistic regime and should also be applicable here.

Figure 1:

Phase diagram depicting the escape regimes of the biased random walk on the subcritical GW tree for different combinations of mean and bias. In Region A the walk is recurrent. In Region B the walk is ballistic and satisfies a functional central limit theorem. In Region C the walk is ballistic and has fluctuations of order $n^{1/\gamma}$. In Region D the walk is sub-ballistic, and the distance reached by the walker by time n is of order $n^\gamma$ (meaning level n is first reached around time $n^{1/\gamma}$).

The case in which $\beta>\mu^{-1}$ has been studied further in [Reference Bowditch15], where it is shown that $|X_n|$ scales with $n^\gamma$ but also experiences the lattice effect. This corresponds to the regime covered by Theorem 1.1, which we do not study further here. (See Figure 1 for a phase diagram depicting the scaling regimes for a biased random walk on a supercritical GW tree.) Furthermore, results similar to those mentioned above have been shown for the random walk on critical and supercritical GW trees [Reference Adékon1, Reference Ben Arous, Fribergh, Gantert and Hammond7, Reference Bowditch16, Reference Croydon, Fribergh and Kumagai19, Reference Lyons, Pemantle and Peres37, Reference Peres and Zeitouni41]. We also refer the reader to [Reference Lyons and Peres38] for further background on random walks on GW trees and their connection to electrical network theory.

Although Theorem 1.3 does not extend to random walks on supercritical GW trees, many of the techniques used in the proofs of Theorems 1.2 and 1.3 can be adapted to the supercritical setting. In particular, a corresponding result should hold with $\beta_c^{1/2}<\beta<\beta_c$ and $\gamma=\log(\beta_c)/\log(\beta)$, where $\beta_c$ is the critical value of the bias determined in [Reference Lyons, Pemantle and Peres37]. Proving this would confirm [Reference Ben Arous and Fribergh6, Conjecture 3.3].

The remainder of this section will proceed as follows. In Subsection 5.1 we show that we can consider the walk on the tree in the RTRW framework. This is done by extending the GW tree conditioned to survive into a two-sided tree, constructing holding times for an RTRW using a sequence of GW trees, and showing that the finite trees are sufficiently short so that the walk never deviates too far from the backbone. In Subsection 5.2 we decompose the trapping times and use Corollary 3.2 to remove parts of the excursions which do not contribute to the limiting process. Finally, in Subsection 5.3, we show that the asymptotic (1.4) holds with these reduced excursion times. This allows us to deduce the result from Theorem 1.2.

5.1. The walk on the tree as a randomly trapped random walk

In this section we show that it suffices to consider a certain RTRW instead of the walk on a GW tree. We begin by extending the GW tree to a two-sided version.

A two-sided tree ${\mathcal{T}}^*$ can be constructed as the extension of a subcritical GW tree conditioned to survive by using the infinite backbone ${\mathcal{Y}}^*=(\ldots,\rho_{-1},\rho_0,\rho_1,\ldots)$ and i.i.d. branches ${\mathcal{T}}^{*-}_k$. That is, ${\mathcal{T}}^{*-}_k$ is the tree formed by the descendants of $\rho_k$ that are not in the descendant tree of $\rho_{k+1}$ (see Figure 2). The original tree ${\mathcal{T}}$ is then the maximal subtree of ${\mathcal{T}}^*$ rooted at $\rho_0$. Let $X^*$ be a random walk on ${\mathcal{T}}^*$ and X the walk on ${\mathcal{T}}$ coupled to $X^*$ so that

\begin{equation*}X_n=X^*_{{\mathscr{L}}(n)} \quad \text{where} \quad {\mathscr{L}}(n)=\min\left\{m\geq 0\;:\;\sum_{k=1}^m{\textbf{1}}_{\{X^*_{k-1},X^*_k\in{\mathcal{T}}\}}\geq n\right\}\!.\end{equation*}

The process X is then a biased random walk on ${\mathcal{T}}$ whose transitions coincide with the transitions of $X^*$ restricted to ${\mathcal{T}}$. The two walks X and $X^*$ deviate by at most the time spent by $X^*$ in ${\mathcal{T}}^*\setminus{\mathcal{T}}$. By transience of the embedded walk on ${\mathcal{Y}}^*$, the walk $X^*$ a.s. spends a finite amount of time outside the infinite subtree rooted at $\rho_0$ (cf. [Reference Bowditch14, Section 2.5]). In particular, $d(X_n,X^*_n)<\log(n)$ with high probability; therefore, since we consider polynomial scaling, we can consider the walk $X^*$.

Figure 2:

Right: a sample two-sided tree ${\mathcal{T}}^*$ with the backbone ${\mathcal{Y}}^*$ represented by solid lines and the buds and traps connected by dashed lines. Left: the subcritical GW tree conditioned to survive ${\mathcal{T}}$ associated with ${\mathcal{T}}^*$.

We now construct the holding times of the RTRW via a sequence of i.i.d. trees. For $k\in{\mathbb{Z}}$, denote by ${{\overline{{\mathcal{T}}}}}_k$ the finite tree ${\mathcal{T}}^{*-}_k$ with an additional vertex ${{\overline{\rho}}}_k$ (germ) appended as a parent of the root $\rho_k$. Transitions to this additional vertex will correspond to transitions along the backbone.

Consider a walk W on ${{\overline{{\mathcal{T}}}}}_k$ with transition probabilities

(5.2) \begin{equation}{\textit{P}}^{{{\overline{{\mathcal{T}}}}}_k}(W_{n+1}=y|W_n=w)=\begin{cases}1 & \text{if } w=y={{\overline{\rho}}}_k, \\[4pt] \dfrac{\beta+1}{1+\beta (|c(w)|+1)} & \text{if } w=\rho_k, \ y={{\overline{\rho}}}_k, \\[10pt] \dfrac{\beta}{1+\beta (|c(w)|+1)} & \text{if } w=\rho_k, \ y \in c(w), \\[10pt] \dfrac{\beta}{1+\beta|c(w)|} & \text{if } w \notin\{\rho_k,{{\overline{\rho}}}_k\}, \ y \in c(w), \\[10pt] \dfrac{1}{1+\beta|c(w)|} & \text{if } w \notin\{\rho_k,{{\overline{\rho}}}_k\}, \ y =\overleftarrow{w}, \\[10pt] 0 & \text{otherwise.} \\ \end{cases}\end{equation}

An excursion by W in ${{\overline{{\mathcal{T}}}}}_k$ started from $\rho_k$ until absorption in ${{\overline{\rho}}}_k$ has the same distribution as the time taken to move between backbone vertices of ${\mathcal{T}}^*$ started from $\rho_k$. In particular, we can construct an RTRW on ${\mathbb{Z}}$ with embedded walk given by the projection of $Y^*$ (the embedded walk of $X^*$) onto ${\mathbb{Z}}$, environment determined by $\omega=({{\overline{{\mathcal{T}}}}}_k)_{k \in {\mathbb{Z}}}$, and holding times given by the excursions of $X^*$. In particular, we have that the RTRW is at $k \in {\mathbb{Z}}$ precisely when $X^*$ is in ${\mathcal{T}}^{*-}_k$.

For C large, up to level n there will be no branches of height greater than $C\log(n)$ with high probability (cf. [Reference Bowditch15, Corollary 3.2]). It follows that $X^*$ and the RTRW deviate by at most $C\log(n)$ with high probability. Since we consider polynomial scaling, it suffices to consider this RTRW. To ease notation, we now denote by X the RTRW.

5.2. A decomposition of the trapping times

By Theorem 1.2, in order to prove Theorem 1.3, it suffices to show that for any $\ell \in{\mathbb{N}}$, $\lambda>0$ and a real-valued function f satisfying $f(1,\lambda)\asymp\lambda^{\gamma}$,

\begin{equation*}n_k(\varsigma)\log\left({\textbf{E}}\left[{\textit{E}}^\omega\left[\exp\left(-\frac{\lambda}{n_k(\varsigma)^{1/\gamma}}(\eta_0-{\mathbb{E}}[\eta_0])\right)\right]^\ell \right]\right)\sim f(\ell ,\lambda)\end{equation*}

as $k \rightarrow \infty$, where we recall that $\gamma=-\log(\mu)/\log(\beta)\in(1,2)$. With a slight abuse of notation we will write n instead of $n_k(\varsigma)$ until we need to consider the specific subsequences.

The purpose of this Section is to decompose the holding times of the RTRW; we begin by proving upper bounds on the expected excursion times of biased random walks in trees. Let ${\mathscr{T}}$ be a fixed tree rooted at a vertex $\rho$ with germ ${{\overline{\rho}}}$. Let W be a random walk on ${\mathscr{T}}$ with transitions given by (5.2) and first return times $\tau^+_x$. Let ${{\overline{{\mathcal{T}}}}}$ be a random tree with germ ${{\overline{\rho}}}$, root $\rho$, and first generation size $\xi^*-1$, which are the roots of independent GW trees $({\mathcal{T}}^\xi_k)_{k=1}^{\xi^*-1}$. Let ${{\overline{{\mathcal{H}}}}}$ denote the height of ${{\overline{{\mathcal{T}}}}}$. Note that for Lemma 5.1 we only assume that $\beta>1$ and $\mu<1$, without the additional assumption that $\mu^{-1/2}<\beta<\mu^{-1}$ that we impose for Theorem 1.3.

Lemma 5.1. Suppose that ${\textbf{E}}[\xi^4]<\infty$, $\beta>1$, and $\mu<1$; then for any $\alpha\in(1,2]$ there exist constants $C_1, C_2<\infty$ such that for any $h\geq 1$ the following hold:

1. 1. ${\textit{E}}^{{\mathscr{T}}}_\rho[(\tau^+_{{{\overline{\rho}}}})^\alpha] \leq C_1\sum\limits_{k=0}^\infty \sum\limits_{j=0}^{\infty}Z_k^{{\mathscr{T}}}(Z_j^{{\mathscr{T}}})^2\beta^{j\alpha}; $

2. 2. ${\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}_\rho[(\tau^+_{{{\overline{\rho}}}})^\alpha]\big|{{\overline{{\mathcal{H}}}}} \leq h\right]\leq C_2h(\beta^\alpha\mu)^h.$

Proof. For x in ${\mathscr{T}}$ let $v_x$ denote the number of visits to x before reaching ${{\overline{\rho}}}$. Using convexity and the fact that $v_x$ is geometrically distributed, we have that ${\textit{E}}^{{\mathscr{T}}}_\rho[(\tau^+_{{{\overline{\rho}}}})^\alpha] $ is equal to

\begin{equation*} {\textit{E}}^{{\mathscr{T}}}_\rho\left[\left(\sum_{x\in{\mathscr{T}}}v_x\right)^\alpha\right] \leq \left(\sum_{x\in{\mathscr{T}}}1\right)\left(\sum_{x\in{\mathscr{T}}}{\textit{E}}^{{\mathscr{T}}}_\rho\left[v_x^\alpha\right]\right) \leq C\sum_{k=0}^\infty Z_k^{{\mathscr{T}}}\sum_{x\in{\mathscr{T}}}(\beta |c(x)|+1)^\alpha\beta^{\alpha|\rho-x|}.\end{equation*}

Noting that $(\beta |c(x)|+1)^\alpha\leq (\beta |c(x)|+1)^2\leq C((Z_{|x|+1}^{{\mathscr{T}}})^2+(Z_{|x|}^{{\mathscr{T}}})^2)$ gives the first result.

By [Reference Bowditch17, Lemma 4.1],

\[{\textbf{E}}\left[Z_{k}^{{\mathcal{T}}^\xi_1}Z_{j}^{{\mathcal{T}}^\xi_1}Z_{i}^{{\mathcal{T}}^\xi_1}\right]\leq C\mu^{k\lor j \lor i}\]

for any i,j,k. We therefore have that ${\textbf{E}}[Z_k^{{{\overline{{\mathcal{T}}}}}}(Z_j^{{{\overline{{\mathcal{T}}}}}})^2|{{\overline{{\mathcal{H}}}}}\leq h]$ is bounded above by

\begin{equation*} \frac{{\textbf{1}}_{\{j,k\leq h\}}{\textbf{E}}[Z_k^{{{\overline{{\mathcal{T}}}}}}(Z_j^{{{\overline{{\mathcal{T}}}}}})^2]}{{\mathbb{P}}({{\overline{{\mathcal{H}}}}}\leq h)}\leq C{\textbf{1}}_{\{j,k\leq h\}}{\textbf{E}}[(\xi^*)^3]{\textbf{E}}\left[Z_{k-1}^{{\mathcal{T}}^\xi_1}\left(Z_{j-1}^{{\mathcal{T}}^\xi_1}\right)^2\right]\leq C\mu^{k\lor j}{\textbf{1}}_{\{j,k\leq h\}},\end{equation*}

since ${\mathbb{P}}({{\overline{{\mathcal{H}}}}}\leq 1)$ is bounded below by a constant and ${\textbf{E}}[\xi^4]<\infty$ implies that ${\textbf{E}}[(\xi^*)^3]<\infty$. Substituting this into the previous equation gives the second result.□

Since we can consider a single excursion in an individual branch, we will simplify notation by studying the holding time $\eta_0$ as the excursion time of the random walk W (whose transition probabilities are given in (5.2)) on the dummy tree ${{\overline{{\mathcal{T}}}}}$ with germ ${{\overline{\rho}}}$ and root $\rho$ which has $\xi^*-1$ offspring $(\rho_k)_{k=1}^{\xi^*-1}$ which are roots of independent GW trees $({\mathcal{T}}^\xi_k)_{k=1}^{\xi^*-1}$. We write ${{\overline{{\mathcal{T}}}}}^\xi_k$ for the tree formed by attaching the root $\rho$ to the tree ${\mathcal{T}}^\xi_k$. Let

\begin{equation*}N_k\,{:\!=}\sum_{n=1}^{\eta_0}{\textbf{1}}_{\{W_{n-1}=\rho,\;W_n=\rho_k\}}\end{equation*}

be the number of entrances into the tree ${\mathcal{T}}^\xi_k$. Letting $\tau^{0,k}=-1$, we have that $\tau^{j,k}\,{:\!=}\inf\{n>\tau^{j-1,k}:W_n=\rho, \; W_{n+1}=\rho_k\}$ is the start time of the jth excursion into ${\mathcal{T}}^\xi_k$, and $\tau^{j,k}_\rho\,{:\!=}\inf\{n>\tau^{j,k}:W_n=\rho\}$ is the end time of the excursion. It follows that

\begin{equation*}\eta_0=1+\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k}\big(\tau^{j,k}_\rho-\tau^{j,k}\big),\end{equation*}

where the 1 corresponds to the final step from the root to the germ. Each trap ${\mathcal{T}}^\xi_k$ has a deepest vertex which we denote by $\delta_k$ (if there is more than one of equal depth then we choose one from the final generation uniformly at random). We can then write

\begin{equation*}G_{j,k}\,{:\!=}\sum_{n=\tau^{j,k}}^{\tau^{j,k}_\rho}{\textbf{1}}_{\{W_n=\delta_k\}}\end{equation*}

for the number of visits to $\delta_k$ on the jth excursion into ${\mathcal{T}}^\xi_k$. In particular, we write

1. (i) $\tau^{0,j,k}_\delta\,{:\!=}\,0$ and $\tau^{i,j,k}_\delta\,{:\!=}\inf\{n>\tau^{i-1,j,k}_\delta:W_n=\delta_k\}$ for the hitting times of $\delta_k$ on the jth excursion into ${\mathcal{T}}^\xi_k$;

2. (ii) $T_{i,j,k}\,{:\!=}\, \tau^{i+1,j,k}_\delta-\tau^{i,j,k}_\delta$ for the duration of the ith excursion from $\delta_k$ to itself on the jth excursion into ${\mathcal{T}}^\xi_k$ for $i=1,\ldots,G_{j,k}-1$;

3. (iii)

   \[T_{j,k}\,{:\!=}\, \left(\tau^{j,k}_\rho-\tau^{G_{j,k}-1,j,k}_\delta+\tau^{1,j,k}_\delta-\tau^{j,k}\right){\textbf{1}}_{\{G_{j,k}>0\}}+\left(\tau^{j,k}_\rho-\tau^{j,k}\right){\textbf{1}}_{\{G_{j,k}=0\}}\]
   for the remainder of the jth excursion into ${\mathcal{T}}^\xi_k$ (which consists of the first journey to the deepest point, the last journey from the deepest point, and the entire excursion if the deepest point is not reached).

Using the convention that $\sum_{i=1}^{-1}\lambda_i=0$, we then have

\begin{equation*}\eta_0= 1+\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k} \left(T_{j,k}+\sum_{i=1}^{G_{j,k}-1}T_{i,j,k}\right)\!.\end{equation*}

We now show that we can ignore the parts of the excursion given by $T_{j,k}$, which allows us to consider holding times of the form given in (5.3) below. The proof of this follows from comparing the excursions forming $T_{j,k}$ with excursion times in unconditioned GW trees in which the bias along the spine (\,from the root to the deepest point) acts towards the root and the bias inside the foliage acts away from the spine (but we control the height of the foliage).

Lemma 5.2. Under the assumptions of Theorem 1.3, there exists $\alpha>\gamma$ such that

\begin{equation*}{\mathbb{E}}\left[\left(1+\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k}T_{j,k}\right)^\alpha\right]<\infty.\end{equation*}

Proof. Let

\[T^{(1)}\,{:\!=}\, \big(\tau^{1,1}_\rho-\tau^{G_{1,1}-1,1,1}_\delta\big){\textbf{1}}_{\{G_{1,1}>0\}}\]

be the time taken to return to the root from the deepest point,

\[T^{(2)}\,{:\!=}\, \big(\tau^{1,1,1}_\delta-\tau^{1,1}\big){\textbf{1}}_{\{G_{1,1}>0\}}\]

the time taken to reach the deepest point from the root, and

\[T^{(3)}\,{:\!=}\, \big(\tau^{1,1}_\rho-\tau^{1,1}\big){\textbf{1}}_{\{G_{1,1}=0\}}\]

the duration of an excursion when it does not reach the deepest point. Then $T_{1,1}=T^{(1)}+T^{(2)}+T^{(3)}$. By convexity we have that

\begin{equation*}{\mathbb{E}}\left[\left(1+\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k}T_{j,k}\right)^\alpha\right]\leq C{\mathbb{E}}\left[(\xi^*)^{\alpha}\right]{\mathbb{E}}\left[N_1^\alpha\right]\left({\mathbb{E}}\left[(T^{(1)})^\alpha\right]+{\mathbb{E}}\left[(T^{(2)})^\alpha\right]+{\mathbb{E}}\left[(T^{(3)})^\alpha\right]\right)\!.\end{equation*}

Since $\xi$ has finite third moments we have that $\xi^*$ has finite second moments and therefore ${\mathbb{E}}\left[(\xi^*)^{\alpha}\right]<\infty$ for $\alpha\leq 2$. The number of entrances into the first trap is geometrically distributed with success probability $(\beta+1)/(2\beta+1)$, and therefore ${\mathbb{E}}\left[N_1^\alpha\right]<\infty$ for any $\alpha<\infty$.

For convenience, we will consider a biased walk W on a dummy trap ${\mathcal{T}}^\xi$ with root $\rho$ and deepest point $\delta$. Let ${\mathcal{H}}$ denote the height of ${\mathcal{T}}^\xi$ and ${\mathbb{P}}^h(\!\cdot\!)={\mathbb{P}}(\cdot|{\mathcal{H}}=h)$ the environment law conditioned on the height of the tree. Then

\begin{equation*}{\mathbb{E}}\left[(T^{(1)})^\alpha\right]\leq\sum_{h=1}^\infty {\mathbb{P}}({\mathcal{H}}=h){\textbf{E}}^h\left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta}\left[(\tau_\rho^+)^\alpha|\tau_\rho^+<\tau_\delta^+\right]\right]\!.\end{equation*}

Recall that a trap ${\mathcal{T}}^\xi$ has a unique self-avoiding path connecting $\rho$ and $\delta$, which we call the spine and denote by $\delta^0=\delta,\delta^1, \ldots,\delta^{\mathcal{H}}=\rho$. For a vertex $\delta^k$ on the spine, we refer to the vertices which are descendants of $\delta^k$ but not on the spine as being in the subtrap ${\mathcal{T}}^\xi_{\delta^k}$ at $\delta^k$. Conditioning on $\tau_\rho^+<\tau_\delta^+$ or $\tau_\delta^+<\tau_\rho^+$ does not change the transition probabilities inside the subtraps or the probability of moving into a subtrap (cf. [Reference Ben Arous, Fribergh, Gantert and Hammond7, Section 8]). Let

\begin{equation*}{\mathcal{L}}_k=\sum_{n=1}^{\tau_\delta^+\land\tau_\rho^+}\left({\textbf{1}}_{\{W_{n-1}=\delta^{k-1}\}\cap\{W_n=\delta^k\}}+{\textbf{1}}_{\{W_{n-1}=\delta^{k+1}\}\cap\{W_n=\delta^k\}}\right)\end{equation*}

denote the number of visits to $\delta^k$ from one of its neighbours on the spine before reaching $\rho$ or $\delta$. Using convexity we then have that

\begin{equation*}{\mathbb{E}} \left[(T^{(1)})^\alpha\right] \leq \sum_{h=1}^\infty {\mathbb{P}}({\mathcal{H}}=h)h^{\alpha-1} \sum_{k=1}^h {\textbf{E}}^h \left[ {\textit{E}}^{{\mathcal{T}}^\xi}_{\delta} \left[{\mathcal{L}}_k^\alpha|\tau_\rho^+<\tau_\delta^+\right]\right] {\textbf{E}}^h \left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta^k}\left[(\tau^+_{\delta^{k-1}}\land\tau^+_{\delta^{k+1}})^\alpha\right]\right] .\end{equation*}

Similarly, we have

\begin{align*}{\mathbb{E}} \left[(T^{(2)})^\alpha\right]& \leq \sum_{h=1}^\infty {\mathbb{P}}({\mathcal{H}}=h)h^{\alpha-1} \sum_{k=1}^h {\textbf{E}}^h \left[ {\textit{E}}^{{\mathcal{T}}^\xi}_{\rho} \left[{\mathcal{L}}_k^\alpha|\tau_\delta^+<\tau_\rho^+\right]\right] {\textbf{E}}^h \left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta^k}\left[(\tau^+_{\delta^{k-1}}\land\tau^+_{\delta^{k+1}})^\alpha\right]\right]\!, \\{\mathbb{E}} \left[(T^{(3)})^\alpha\right]& \leq \sum_{h=1}^\infty {\mathbb{P}}({\mathcal{H}}=h)h^{\alpha-1} \sum_{k=1}^h {\textbf{E}}^h \left[ {\textit{E}}^{{\mathcal{T}}^\xi}_{\rho} \left[{\mathcal{L}}_k^\alpha|\tau_\rho^+<\tau_\delta^+\right]\right] {\textbf{E}}^h \left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta^k}\left[(\tau^+_{\delta^{k-1}}\land\tau^+_{\delta^{k+1}})^\alpha\right]\right]\!.\end{align*}

The only term that differs in these expressions is the expected local time at vertices on the spine. By the gambler’s ruin, the walk on the spine can be stochastically dominated by a biased random walk where the bias favours steps towards $\delta$ when conditioned on $\tau_\delta^+<\tau_\rho^+$ and towards $\rho$ when conditioned on $\tau_\rho^+<\tau_\delta^+$. It immediately follows that there exists a constant C which is independent of h and k such that

\begin{equation*}{\textbf{E}}^h\left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta}\left[{\mathcal{L}}_k^\alpha|\tau_\rho^+<\tau_\delta^+\right]\right]\!,{\textbf{E}}^h\left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\rho}\left[{\mathcal{L}}_k^\alpha|\tau_\delta^+<\tau_\rho^+\right]\right]\!,{\textbf{E}}^h\left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\rho}\left[{\mathcal{L}}_k^\alpha|\tau_\rho^+<\tau_\delta^+\right]\right]\leq C.\end{equation*}

Since the number of children $\delta^k$ has off the spine is stochastically dominated by a size-biased random variable and each of these is the root of a GW tree conditioned to have height at most $k-1$, by Lemma 5.1,

\begin{equation*}{\textbf{E}}^h\left[{\textit{E}}^{{\mathcal{T}}^\xi}_{\delta^k}\left[(\tau^+_{\delta^{k-1}}\land\tau^+_{\delta^{k+1}})^\alpha\right]\right]\leq{\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}_\rho[(\tau^+_{{{\overline{\rho}}}})^\alpha]\big|{{\overline{{\mathcal{H}}}}}\leq k\right] \leq Ck(\beta^\alpha\mu)^{k}. \end{equation*}

By (5.1) we have that ${\mathbb{P}}({\mathcal{H}}=h)\leq C\mu^h$ for some constant $C<\infty$. It follows that for $i=1,2,3$,

\begin{equation*}{\mathbb{E}}\left[(T^{(i)})^\alpha\right]\leq C\sum_{h=1}^\infty \mu^hh^{\alpha-1}\sum_{k=1}^hk(\beta^\alpha\mu)^k \leq C\sum_{h=1}^\infty h^{\alpha}(\beta^\alpha\mu^2)^h,\end{equation*}

which is finite since $\beta^\alpha \mu^2<1$.□

By Lemma 5.2 and Corollary 3.2 we can now consider holding times of the form

(5.3) \begin{equation}\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k} \sum_{i=1}^{G_{j,k}-1}T_{i,j,k}.\end{equation}

The excursions $T_{i,j,k}$ from the deepest point $\delta_k$ in the trap ${\mathcal{T}}^\xi_k$ are typically very short because the bias is acting towards $\delta_k$. They do, however, depend on the height of the trap, which (through $G_{j,k}$) is largely responsible for the heavy-tailed behaviour in the above sum. We therefore want to consider excursion times from the deepest point that do not depend on the height of the trap; for this, we will replace $T_{i,j,k}$ with excursion times on ${\mathcal{T}}^\xi_k$ extended to an infinite trap ${\mathcal{T}}_k^*$ using the construction of [Reference Geiger and Kersting28]. That is, starting from ${\mathcal{T}}^\xi_k$, we can iteratively construct a random infinite tree ${\mathcal{T}}_k^*$ with deepest point $\delta_k$ such that, denoting by $\delta_k^1$ the parent of $\delta_k$ and by $\delta_k^{m+1}$ the parent of $\delta_k^m$, the following hold:

1. 1. The subtree of ${\mathcal{T}}_k^*$ consisting of $\delta_k^m $ and all of its descendants is a GW tree conditioned to have height m.

2. 2. For $m =|\rho_k-\delta_k|$, the subtree of ${\mathcal{T}}_k^*$ consisting of $\delta_k^m $ and all of its descendants is ${\mathcal{T}}^\xi_k$.

We now show that we can replace $T_{i,j,k}$ with excursion times on ${\mathcal{T}}^*_k$. We begin by constructing these excursion times. For $k= 1,\ldots,\xi^*-1$ we make the following definitions:

1. 1. Let $X_n^k$ be independent $\beta$-biased random walks on ${\mathcal{T}}_k^*$ started from $X_0^k=\delta_k$.

2. 2. Let $T_{1,1,k}^*=\inf\{n>0:X_n^k=\delta_k\}$ be the first return time to $\delta_k$.

3. 3. Let ${\mathcal{A}}_{1,1,k}\,{:\!=}\{T_{1,1,k}^*>\inf\{n>0:X_n^k=\delta_k^{|\rho_k-\delta_k|}\}\}$ be the event that the walk leaves ${\mathcal{T}}^\xi_k$ before returning to $\delta_k$.

We then have the following:

1. 1. $\tilde{T}_{1,1,k}\,{:\!=}\,T_{1,1,k}^*{\textbf{1}}_{{\mathcal{A}}_{1,1,k}}+T_{1,1,k}{\textbf{1}}_{{\mathcal{A}}_{1,1,k}^c}$ has the same law as $T_{1,1,k}^*$ but is coupled to $T_{1,1,k}$ such that $T_{1,1,k}=\tilde{T}_{1,1,k}+{\textbf{1}}_{{\mathcal{A}}_{1,1,k}}(T_{1,1,k}-T_{1,1,k}^*)$.

2. 2. $(\tilde{T}_{1,1,k})_{k\geq 1}$ are i.i.d. with respect to ${\mathbb{P}}$.

3. 3. For each k we can construct $(\tilde{T}_{i,j,k})_{i,j\geq 1}$ which are

   1. (i) i.i.d. with respect to ${\textit{P}}^{{\mathcal{T}}^*_k}$;

   2. (ii) coupled to $T_{i,j,k}$ in such a way that $T_{i,j,k}=\tilde{T}_{i,j,k}+{\textbf{1}}_{{\mathcal{A}}_{i,j,k}}(T_{i,j,k}-T_{i,j,k}^*)$, where $(T_{i,j,k}^*)_{i,j\geq 1}$ are independent excursion times on ${\mathcal{T}}_k^*$, and ${\mathcal{A}}_{i,j,k}$ are the events that the corresponding walks leave ${\mathcal{T}}_k^\xi$.

Before showing that we can replace $T_{i,j,k}$ with $\tilde{T}_{i,j,k}$, we first show that the $\tilde{T}_{i,j,k}$ have finite second moments.

Lemma 5.3. Under the assumptions of Theorem 1.3 we have that ${\mathbb{E}}[\tilde{T}_{1,1,1}^2]<\infty$.

Proof. Let $\delta^*$ denote the spine $\{\delta^0,\delta^1,\ldots\}$ of ${\mathcal{T}}^*$, and for $x\in{\mathcal{T}}^*$ let $v_x$ denote the number of visits to x before returning to $\delta=\delta^0$. By the gambler’s ruin we have that ${\textit{P}}^{{\mathcal{T}}^*}_\delta(\tau_{\delta^k}^+<\tau_{\delta}^+)\leq C\beta^{-k}$. Since the walk on the spine is $\beta$-biased we have that ${\textit{E}}^{\delta^*}_{\delta^k}[(v_{\delta^k})^2]\leq C$ independently of k. Moreover, the time spent in a subtrap rooted at $\delta^k$ is stochastically dominated by the time spent in the tree ${{\overline{{\mathcal{T}}}}}$ conditioned to have height at most k. By Lemma 5.1 it follows that

\begin{align*}{\mathbb{E}}[\tilde{T}_{1,1,1}^2] &\leq \sum_{k=0}^\infty {\textbf{E}}\left[{\textit{P}}^{{\mathcal{T}}^*}_\delta(\tau_{\delta^k}^+<\tau_{\delta}^+)\right]{\textbf{E}}\left[{\textit{E}}^{\delta^*}_{\delta^k}[(v_{\delta^k})^2]\right]{\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}_\rho[(\tau^+_{{{\overline{\rho}}}})^2]|{{\overline{{\mathcal{H}}}}}\leq k\right] \\ &\leq \sum_{k=0}^\infty\beta^{-k}k(\beta^2\mu)^k,\end{align*}

which is finite since $\beta\mu<1$.□

The following lemma shows that we can replace the excursions $T_{i,j,k}$ with excursions on the infinite traps (which are independent of the height of the original trap).

Lemma 5.4. Under the assumptions of Theorem 1.3 we have that, for some $\alpha>\gamma$,

\begin{equation*}{\mathbb{E}}\left[\left(\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k}\sum_{i=1}^{G_{j,k}-1}{\textbf{1}}_{{\mathcal{A}}_{i,j,k}}(T_{i,j,k}-T_{i,j,k}^*)\right)^{ \alpha}\right]<\infty.\end{equation*}

Proof. Since ${\textbf{E}}[(\xi^*)^\alpha],{\textbf{E}}[N_k^\alpha]<\infty$, by convexity it suffices to show that

(5.4) \begin{equation}{\mathbb{E}}\left[\left(\sum_{i=1}^{G_{1,1}-1}{\textbf{1}}_{{\mathcal{A}}_{i,1,1}}(T_{i,1,1}-T_{i,1,1}^*)\right)^{ \alpha}\right]\end{equation}

is finite. Summing over the height ${\mathcal{H}}$ of the trap ${\mathcal{T}}^\xi_1$ and using that ${\mathbb{P}}({\mathcal{H}}=h)\leq C\mu^h$, we have that (5.4) is bounded above by

\begin{align*} &\sum_{h=1}^\infty {\mathbb{P}}({\mathcal{H}}=h){\mathbb{E}}^h \left[ \left(\sum_{i=1}^{G_{1,1}-1}{\textbf{1}}_{{\mathcal{A}}_{i,1,1}}(T_{i,1,1}-T_{i,1,1}^*)\right)^{ \alpha}\right] \\ & \qquad \leq \sum_{h=1}^\infty \mu^h{\mathbb{E}}^h \left[G_{1,1}^\alpha\right] {\mathbb{E}}^h \left[\left({\textbf{1}}_{\{{\mathcal{A}}_{1,1,1}\}}(T_{1,1,1}-T_{1,1,1}^*)\right)^{\alpha}\right]\!.\end{align*}

It follows from Lemma 5.1 that ${\mathbb{E}}\left[\left(T_{1,1,1}-T_{1,1,1}^*\right)^\alpha\big| {\mathcal{A}}_{1,1,1}, {\mathcal{H}}=h\right]\leq h(\beta^\alpha\mu)^h$. Conditional on ${\mathcal{H}}=h$ we have that $G_{1,1}$ is geometrically distributed with success probability $(\beta^h-\beta)/(\beta^h-1)$, which gives us that ${\mathbb{E}}\left[G_{1,1}^\alpha\big| {\mathcal{H}}=h\right]\leq C\beta^{\alpha h}$. Similarly, ${\mathbb{P}}({\mathcal{A}}_{1,1,1}|{\mathcal{H}}=h)\leq C\beta^{-h}$, and therefore we have that (5.4) is bounded above by

\begin{equation*}\sum_{h=1}^\infty \mu^h\beta^{h(\alpha-1)} {\mathbb{E}}\left[\left(T_{1,1,1}-T_{1,1,1}^*\right)^\alpha\big| {\mathcal{A}}_{1,1,1}, {\mathcal{H}}=h\right]\leq \sum_{h=1}^\infty h(\beta^{2\alpha-1}\mu^2)^h.\end{equation*}

Since $\alpha>\gamma=\log(\mu^{-1})/\log(\beta)$, we have that $\beta^{2\alpha-1}\mu^2=\beta^{2(\alpha-\gamma)-1}<1$ for $\alpha<\gamma+1/2$. For such $\alpha$ we then have that the above sum is finite, which completes the proof.□

By Lemma 5.4 and Corollary 3.2 we can now consider holding times of the form

\begin{equation*}\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{N_k} \sum_{i=1}^{G_{j,k}-1}\tilde{T}_{i,j,k}.\end{equation*}

On the event that $\delta_k$ is reached on the jth excursion into ${\mathcal{T}}_k^\xi$, the random variable $G_{j,k}-1$ is geometrically distributed with success probability

\begin{equation*}q_{{\mathcal{H}}_k}\,{:\!=}\,{\textit{P}}_{\delta_k}^{{\mathcal{T}}^\xi_k}(\tau_\rho^+<\tau_{\delta_k}^+)=\frac{\beta-1}{\beta^{{\mathcal{H}}_k}-1},\end{equation*}

where ${\mathcal{H}}_k=|\rho-\delta_k|$ is the height of the trap ${\mathcal{T}}_k^\xi$. We will replace $N_k$ with the number of excursions into ${\mathcal{T}}_k^\xi$ such that $\delta_k$ is reached, so that we can consider $G_{j,k}$ as geometrically distributed. The probability that $\delta_k$ is reached on a given excursion depends on the height of the trap; we also remove this dependency.

Write

\[B_k\,{:\!=}\sum_{j=1}^{N_k}{\textbf{1}}_{\{G_{j,k}>0\}}\]

for the number of excursions that reach $\delta_k$; then $B_k$ is binomially distributed with $N_k$ trials and success probability

\begin{equation*}{\textit{P}}_{\rho_k}^{{{\overline{{\mathcal{T}}}}}_k^\xi}(\tau_{\delta_k}^+<\tau_{\rho}^+)=\frac{\beta^{{\mathcal{H}}_k}(1-\beta^{-1})}{\beta^{{\mathcal{H}}_k}-1}=(1-\beta^{-1})+\frac{1-\beta^{-1}}{\beta^{{\mathcal{H}}_k}-1}.\end{equation*}

For $B_k$ fixed, let $\tilde{B}_k$ be binomially distributed with $B_k$ trials and success probability $1-\beta^{-{\mathcal{H}}_k}$. Then $\tilde{B}_k$ is coupled to $B_k$ so that $\tilde{B}_k\leq B_k$ but has the distribution of a binomial random variable with $N_k$ trials and success probability $1-\beta^{-1}$. Let $\tilde{G}_{j,k}$ be independent and equal in distribution (with respect to ${\textit{P}}^{{{\overline{{\mathcal{T}}}}}}$) to $G_{j,k}-1$ conditional on $\{G_{j,k}>0\}$.

Lemma 5.5. Under the assumptions of Theorem 1.3 we have that, for some $\alpha>\gamma$,

\begin{equation*}{\mathbb{E}}\left[\left(\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{B_k-\tilde{B}_k}\sum_{i=1}^{\tilde{G}_{j,k}}\tilde{T}_{i,j,k}\right)^{ \alpha}\right]<\infty.\end{equation*}

Proof. We can choose $\alpha>\gamma$ such that ${\textbf{E}}[(\xi^*)^\alpha]<\infty$; therefore, by convexity, it suffices to show that

(5.5) \begin{equation}{\mathbb{E}}\left[\left(\sum_{j=1}^{B_1-\tilde{B}_1}\sum_{i=1}^{\tilde{G}_{j,1}}\tilde{T}_{i,j,1}\right)^{ \alpha}\right]\end{equation}

in finite.

Since $B_1-\tilde{B}_1\leq N_1{\textbf{1}}_{\{B_1\neq \tilde{B_1}\}}$ we have that (5.5) is bounded above by

\begin{equation*}{\mathbb{E}}\left[\left(\sum_{j=1}^{N_1}{\textbf{1}}_{\{B_1\neq \tilde{B_1}\}}\sum_{i=1}^{\tilde{G}_{j,1}}\tilde{T}_{i,j,1}\right)^{ \alpha}\right] \leq {\mathbb{E}}[N_1^\alpha]{\mathbb{E}}\left[\left({\textbf{1}}_{\{B_1\neq \tilde{B_1}\}}\sum_{i=1}^{\tilde{G}_{1,1}}\tilde{T}_{i,1,1}\right)^{ \alpha}\right]\!,\end{equation*}

where we have that ${\mathbb{E}}[N_1^\alpha]<\infty$. Summing over the height ${\mathcal{H}}$ of the trap ${\mathcal{T}}^\xi_1$, using the fact that ${\mathbb{P}}({\mathcal{H}}=h)\leq C\mu^h$ and the independence of $\tilde{G}_{1,1}$ with $(\tilde{T}_{i,1,1})_{i\geq 1}$, we have that (5.5) is bounded above by

\begin{equation*} C\sum_{h=1}^\infty \mu^h {\mathbb{P}}(B_1\neq \tilde{B}_1|{\mathcal{H}}=h){\mathbb{E}}[\tilde{G}_{1,1}^\alpha|{\mathcal{H}}=h]{\mathbb{E}}[\tilde{T}_{1,1,1}^\alpha].\end{equation*}

By definition of $B_k$ and $\tilde{B}_k$ we have that ${\mathbb{P}}(B_1\neq \tilde{B}_1|{\mathcal{H}}=h)\leq C \beta^{-h}$, and, as in Lemma 5.4, we have that ${\mathbb{E}}[\tilde{G}_{1,1}^\alpha|{\mathcal{H}}=h]\leq \beta^{\alpha h}$. By Lemma 5.1 we have that ${\mathbb{E}}[\tilde{T}_{1,1,1}^\alpha]<\infty$; therefore it follows that (5.5) can be bounded above by

\begin{equation*}C\sum_{h=1}^\infty (\beta^{\alpha-1}\mu)^h,\end{equation*}

which completes the proof since $\beta^{\alpha-1}\mu \in(0,1)$.□

By Corollary 3.2 we can now consider holding times of the form

\begin{equation*}\tilde{\eta}_0\,{:\!=}\sum_{k=1}^{\xi^*-1}\sum_{j=1}^{\tilde{B}_k} \sum_{i=1}^{\tilde{G}_{j,k}}\tilde{T}_{i,j,k},\end{equation*}

where we recall that

1. (i) $\xi^*-1$ is the number of children of $\rho$ in ${{\overline{{\mathcal{T}}}}}$;

2. (ii) $\tilde{B}_k$ are jointly equal in distribution to the number of excursions into ${\mathcal{T}}_k^\xi$ which reach the deepest points $\delta_k$;

3. (iii) $\tilde{G}_{j,k}$ are equal in distribution to the number of excursions from $\delta_k$ to itself (for a walk started at $\delta_k$) which depend on ${{\overline{{\mathcal{T}}}}}$ only through ${\mathcal{H}}_k$;

4. (iv) $\tilde{T}_{i,j,k}$ are excursions (from $\delta_k$ to itself) in the infinite traps ${\mathcal{T}}_k^*$ and, conditional on ${\mathcal{T}}_k^*$, are independent of ${\mathcal{H}}_k$.

5.3. Subsequential convergence of the Laplace transforms

In this section we show that for any $\ell \in{\mathbb{N}}$, $\lambda>0$ and a real-valued function f satisfying $f(1,\lambda)\asymp\lambda^{\gamma}$,

\begin{equation*}n_k(\varsigma)\log\left({\textbf{E}}\left[{\textit{E}}^{{\overline{{\mathcal{T}}}}}\left[\exp\left(-\frac{\lambda}{n_k(\varsigma)^{1/\gamma}}(\tilde{\eta}_0-{\mathbb{E}}[\tilde{\eta}_0])\right)\right]^\ell \right]\right)\sim\,f(\ell ,\lambda)\end{equation*}

as $k \rightarrow \infty$. As before, we continue to write n for $n_k(\varsigma)$ until we require the specific subsequences. We begin by rearranging the quenched Laplace transform of $\tilde{\eta}_0$ using the distributions and dependence structure of $\xi^*$, $\tilde{B}_k$, $\tilde{G}_{j,k}$, and $\tilde{T}_{i,j,k}$ detailed in the previous section.

Lemma 5.6. Under the assumptions of Theorem 1.3 we have that

\begin{equation*}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{\eta}_0\right)\right]= \frac{1}{1+\frac{\beta-1}{\beta+1}\sum_{k=1}^{\xi^*-1}\left(1-{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\sum_{i=1}^{\tilde{G}_{1,k}}\tilde{T}_{i,1,k}\right)\right]\right)}.\end{equation*}

Proof. Write

\[\vartheta_k\,{:\!=}\,{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\sum_{i=1}^{\tilde{G}_{1,k}}\tilde{T}_{i,1,k}\right)\right]\!;\]

then $\xi^*$ and $\vartheta_k$ are deterministic with respect to ${\textit{P}}^{{{\overline{{\mathcal{T}}}}}}$. Since $\tilde{T}_{i,j,k}$ and $\tilde{G}_{j,k}$ are independent of $\tilde{B}_k$, we have that

\begin{align*}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{\eta}_0\right)\right]&= {\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\prod_{k=1}^{\xi^*-1}\vartheta_k^{\tilde{B}_k}\right] \\ &=\sum_{(b_k)_{k=1}^{\xi^*-1}\in {\mathbb{Z}}_+^{\xi^*-1}}\left(\prod_{k=1}^{\xi^*-1}\vartheta_k^{b_k}\right){\textit{P}}^{{{\overline{{\mathcal{T}}}}}}\left(\bigcap_{k=1}^{\xi^*-1}\tilde{B}_k=b_k\right)\!.\end{align*}

Summing over the total number of excursions $\sum_{k=1}^{\xi^*-1}\tilde{B}_k$, we have that the above expression is equal to

\begin{align*}& \sum_{L=0}^\infty \!\left(\frac{(\xi^*-1)(\beta-1)}{\xi^*(\beta-1)+2}\right)^{L}\! \left(\frac{\beta+1}{\xi^*(\beta-1)+2}\right) \sum_{\substack{(b_k)_{k=1}^{\xi^*-1}\in {\mathbb{Z}}_+^{\xi^*-1}\\ \sum_{k=1}^{\xi^*-1}b_k=L}} (\xi^*-1)^{-L}\binom{L}{b_1,\ldots,b_{\xi^*-1}} \prod_{k=1}^{\xi^*-1}\vartheta_k^{b_k} \\& {\qquad \quad}= \frac{\beta+1}{\xi^*(\beta-1)+2}\sum_{L=0}^\infty \left(\frac{(\beta-1)\sum_{k=1}^{\xi^*-1}\vartheta_k}{\xi^*(\beta-1)+2}\right)^L \\& {\qquad \quad}= \frac{1}{1+\frac{\beta-1}{\beta+1}\sum_{k=1}^{\xi^*-1}(1-\vartheta_k)},\end{align*}

where we have used that $0<\vartheta_k<1$.□

The excursion times $(\tilde{T}_{i,1,k})_{i\geq 1}$ are identically distributed with respect to ${\textit{P}}^{{{\overline{{\mathcal{T}}}}}}$ and independent from $\tilde{G}_{1,k}$; therefore, we obtain

(5.6) \begin{align}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\sum_{i=1}^{\tilde{G}_{1,k}}\tilde{T}_{i,1,k}\right)\right]& = {\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\prod_{i=1}^{\tilde{G}_{1,k}}\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{T}_{i,1,k}\right)\right] \notag \\& = {\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{T}_{1,1,k}\right)\right]^{\tilde{G}_{1,k}}\right] \notag \\& = \frac{1}{1+\frac{1-q_{{\mathcal{H}}_k}}{q_{{\mathcal{H}}_k}}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[1-\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{T}_{1,1,k}\right)\right]} \end{align}

by using the probability generating function of a geometric random variable.

Write

\begin{equation*}\psi_n^{(k)}=\frac{1-q_{{\mathcal{H}}_k}}{q_{{\mathcal{H}}_k}}\frac{\lambda}{n^{1/\gamma}}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,k}]=\frac{\beta^{{\mathcal{H}}_k}-\beta}{\beta-1}\frac{\lambda}{n^{1/\gamma}}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,k}];\end{equation*}

then, using Lemma 5.3 and a Taylor expansion, we have that

\begin{equation*}\frac{1-q_{{\mathcal{H}}_k}}{q_{{\mathcal{H}}_k}}{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[1-\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{T}_{1,1,k}\right)\right]=\psi_n^{(k)}+O(n^{-2/\gamma}).\end{equation*}

Combining this with Lemma 5.6 and (5.6), we have that

\begin{equation*}{\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\left[\exp\left(-\frac{\lambda}{n^{1/\gamma}}\tilde{\eta}_0\right)\right]^\ell \right]={\textbf{E}}\left[\left(\frac{1}{1+\frac{\beta-1}{\beta+1}\sum_{k=1}^{\xi^*-1}\left(1-\frac{1}{1+\psi_n^{(k)}}\right)}\right)^\ell \right]+O(n^{-2/\gamma}).\end{equation*}

This reduces the proof of Theorem 1.3 to showing convergence of

(5.7) \begin{equation}n\log\left({\textbf{E}}\left[\left(\frac{1}{1+\frac{\beta-1}{\beta+1}\sum_{k=1}^{\xi^*-1}\left(1-\frac{1}{1+\psi_n^{(k)}}\right)}\right)^\ell \right]\exp\left(\frac{\ell\lambda{\mathbb{E}}[\tilde{\eta}_0]}{n^{1/\gamma}}\right)\right)\end{equation}

along the given subsequences.

Let

\begin{equation*}\Phi_n^{(M)}\,{:\!=}\frac{\beta-1}{\beta+1}\sum_{k=1}^{M}\left(1-\frac{1}{1+\psi_n^{(k)}}\right);\end{equation*}

then

\[{\textbf{E}}[\Phi_n^{(\xi^*-1)}]=\frac{\lambda{\mathbb{E}}[\tilde{\eta}_0]}{n^{1/\gamma}}+\frac{\beta-1}{\beta+1}{\textbf{E}}[\xi^*-1]{\textbf{E}}\left[\frac{(\psi_n^{(1)})^2}{1+\psi_n^{(1)}}\right]\!,\]

and the first expectation in (5.7) can be written as

\begin{equation*}1-\ell\left({\textbf{E}}[\Phi_n^{(\xi^*-1)}]-{\textbf{E}}\left[\frac{(\Phi_n^{(\xi^*-1)})^2}{1+\Phi_n^{(\xi^*-1)}}\right]\right)+\sum_{j=2}^\ell (\!-1)^j\binom{\ell}{j}{\textbf{E}}\left[\left(\frac{\Phi_n^{(\xi^*-1)}}{1+\Phi_n^{(\xi^*-1)}}\right)^j\right]\!.\end{equation*}

Recall that $\gamma \in(1,2)$. Noting that

\begin{equation*}\exp\left(\frac{\ell\lambda{\mathbb{E}}[\tilde{\eta}_0]}{n^{1/\gamma}}\right)=1+\ell\frac{\lambda{\mathbb{E}}[\tilde{\eta}_0]}{n^{1/\gamma}}+O(n^{-2/\gamma})=1+\ell{\textbf{E}}[\Phi_n^{(\xi^*-1)}]+O(n^{-2/\gamma}),\end{equation*}

it suffices to show convergence of

(5.8) \begin{equation}n{\textbf{E}}\left[\frac{(\psi_n^{(1)})^2}{1+\psi_n^{(1)}}\right]\!, \quad n{\textbf{E}}\left[\frac{(\Phi_n^{(\xi^*-1)})^2}{1+\Phi_n^{(\xi^*-1)}}\right]\!, \quad \text{and} \quad n{\textbf{E}}\left[\left(\frac{\Phi_n^{(\xi^*-1)}}{1+\Phi_n^{(\xi^*-1)}}\right)^j\right]\end{equation}

for every $j\geq 2$. We first prove a uniform upper bound on the tail probabilities of the random variables $\psi_n^{(1)}$ and $\Phi_n^{(M)}$.

Lemma 5.7. Under the assumptions of Theorem 1.3, there exists a constant C such that for any $M\in{\mathbb{N}}$ we have

\begin{equation*}n{\mathbb{P}}(\psi_n^{(1)}>t)\leq Ct^{-\gamma} \quad \text{and} \quad n{\mathbb{P}}(\Phi_n^{(M)}>t)\leq CM^{1+\gamma}t^{-\gamma}\end{equation*}

uniformly over $n\in{\mathbb{N}}$ and $t>0$.

Proof. Using that ${\mathbb{P}}({\mathcal{H}}\geq t)\leq C\mu^t$ for some constant C, we have that

\begin{align*}n{\mathbb{P}}(\psi^{(1)}>t) & \leq n{\mathbb{P}}\left(\beta^{{\mathcal{H}}}>\frac{C_{\lambda,\beta} t n^{1/\gamma}}{{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]}\right) \\&\leq n{\textbf{E}}\left[{\mathbb{P}}\left({\mathcal{H}}>\frac{\log\left(C_{\lambda,\beta} t n^{1/\gamma}\right)-\log\left({\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]\right)}{\log(\beta)}\big| {\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]\right)\right] \\& \leq Cn {\textbf{E}}\left[\mu^{\frac{\log\left(C_{\lambda,\beta} t n^{1/\gamma}\right)-\log\left({\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]\right)}{\log(\beta)}}\right] \\& = Ct^{-\gamma} {\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]^{\gamma}\right]\!,\end{align*}

which is bounded above by $Ct^{-\gamma}$ because ${\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]^2\right]<\infty$ by Lemma 5.3. Noting that $\Phi_n^{(M)}$ is bounded above by $C_\beta\sum_{k=1}^M\psi_n^{(k)}$, a union bound and the above estimate on $n{\mathbb{P}}(\psi^{(1)}>t)$ immediately give the bound for $n{\mathbb{P}}(\Phi_n^{(M)}>t)$.□

Recall that we wish to prove convergence of the three terms in (5.8). Using Lemma 5.7 and integration by parts, we have that

(5.9) \begin{align} n{\textbf{E}}\left[\frac{(\psi_n^{(1)})^2}{1+\psi_n^{(1)}}\right] & =n\int_0^\infty \frac{t(2+t)}{(1+t)^2}{\mathbb{P}}(\psi_n^{(1)}>t){\textrm{d}} t,\\ n{\textbf{E}}\left[\frac{(\Phi_n^{(\xi^*-1)})^2}{1+\Phi_n^{(\xi^*-1)}}\right]& = n\sum_{M=1}^\infty{\mathbb{P}}(\xi^*-1=M){\textbf{E}}\left[\frac{(\Phi_n^{(M)})^2}{1+\Phi_n^{(M)}}\right]\notag\\ & = n\sum_{M=1}^\infty{\mathbb{P}}(\xi^*-1=M)\int_0^\infty \frac{t(2+t)}{(1+t)^2}{\mathbb{P}}\left(\Phi_n^{(M)}>t\right){\textrm{d}} t, \notag\end{align}

\begin{align} n{\textbf{E}}\left[\left(\frac{\Phi_n^{(\xi^*-1)}}{1+\Phi_n^{(\xi^*-1)}}\right)^j\right]& = n\sum_{M=1}^\infty{\mathbb{P}}(\xi^*-1=M){\textbf{E}}\left[\left(\frac{\Phi_n^{(M)}}{1+\Phi_n^{(M)}}\right)^j\right]\notag \\& = n\sum_{M=1}^\infty{\mathbb{P}}(\xi^*-1=M)\int_0^\infty \frac{jt^{j-1}}{(1+t)^{j+1}}{\mathbb{P}}(\Phi_n^{(M)}>t){\textrm{d}} t. \notag\end{align}

Noting that $\xi^*-1$ has finite $1+\gamma$ moments and

\begin{equation*}\int_0^\infty \frac{t(2+t)}{(1+t)^2}t^{-\gamma}{\textrm{d}} t, \; \int_0^\infty \frac{jt^{j-1}}{(1+t)^{j+1}}t^{-\gamma}{\textrm{d}} t <\infty,\end{equation*}

from Lemma 5.7 we have that the integrands in the right-hand side of (5.9) are bounded above by an integrable function. By dominated convergence, it follows that it remains to show convergence of $n{\mathbb{P}}(\psi_n^{(1)}>t)$ and $n{\mathbb{P}}(\Phi_n^{(M)}>t)$ for almost every $t>0$.

Lemma 5.8. Under the assumptions of Theorem 1.3 we have that, for every $M\in{\mathbb{N}}$,

\[n_k(\varsigma){\mathbb{P}}(\psi_{n_k(\varsigma)}^{(1)}>t) \quad \text{and} \quad n_k(\varsigma){\mathbb{P}}(\Phi_{n_k(\varsigma)}^{(M)}>t)\]

converge as $k\rightarrow \infty$ for almost every $t>0$.

Proof. Using that ${\mathbb{P}}({\mathcal{H}}\geq m)\sim c_\mu\mu^m$ and ${\textbf{E}}\left[{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}\big[\tilde{T}_{1,1,1}\big]^\gamma\right]<\infty$, we have that

\begin{align*}n_k{\mathbb{P}}(\psi^{(1)}_{n_k}>t)&= n_k{\mathbb{P}}\left(\beta^{{\mathcal{H}}}>\frac{t n_k^{1/\gamma}(\beta-1)}{\lambda{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]}+\beta\right) \\&= n_k{\textbf{E}}\left[{\mathbb{P}}\left({\mathcal{H}}>\left.\frac{\log\left(\frac{t n_k^{1/\gamma}(\beta-1)}{\lambda{\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]}+\beta\right)}{\log(\beta)}\right| {\textit{E}}^{{{\overline{{\mathcal{T}}}}}}[\tilde{T}_{1,1,1}]\right)\right]\end{align*}

converges for almost every $t>0$, by our choice of subsequence $n_k=\lfloor\varsigma\mu^{-k}\rfloor$.

Let

\[\chi_k^{(i)}= \frac{\psi_{n_k}^{(i)}}{(1+\psi_{n_k}^{(i)})};\]

then by the first part of the lemma we have that $n_k{\mathbb{P}}(\chi_{k}^{(i)}>t)$ converges for almost every $t>0$. In particular, $n_k{\mathbb{P}}(\Phi_{n_k}^{(1)}>t)$ converges for almost every $t>0$. To show that $n_k{\mathbb{P}}(\Phi_{n_k}^{(M)}>t)$ converges for almost every $t>0$ we show that

(5.10) \begin{equation}\lim_{k\rightarrow \infty}n_k\big({\mathbb{P}}(\Phi_{n_k}^{(M)}>t)-M{\mathbb{P}}(\Phi_{n_k}^{(1)}>t)\big)=0.\end{equation}

We show that (5.10) holds for $M=2$; convergence then holds for general M by an inductive argument.

For any $t>0$ we have that

\begin{equation*}{\mathbb{P}}\big(\chi_k^{(1)}+\chi_k^{(2)}>t\big) \geq {\mathbb{P}}\big(\chi_k^{(1)}>t\big)+{\mathbb{P}}\big(\chi_k^{(2)}>t\big)-{\mathbb{P}}\big(\chi_k^{(1)}>t,\chi_k^{(2)}>t\big).\end{equation*}

By independence of $\chi_k^{(1)}$ and $\chi_k^{(2)}$ we have that

\[n_k{\mathbb{P}}\big(\chi_k^{(1)}>t,\chi_k^{(2)}>t\big)=n_k{\mathbb{P}}\big(\chi_k^{(1)}>t\big){\mathbb{P}}\big(\chi_k^{(2)}>t\big)\]

converges to 0 as $k\rightarrow \infty$; thus,

\[\liminf_{k\rightarrow\infty}n_k\big({\mathbb{P}}(\Phi_{n_k}^{(2)}>t)-2{\mathbb{P}}(\Phi_{n_k}^{(1)}>t)\big)\geq 0.\]

Next, we have that for any $\varepsilon>0$,

\begin{align*}&{\mathbb{P}}\big(\chi_k^{(1)}+\chi_k^{(2)}>t\big) \\ &\leq {\mathbb{P}}\big(\chi_k^{(1)}>\varepsilon,\chi_k^{(2)}>\varepsilon\big)+{\mathbb{P}}\big(\chi_k^{(1)}>t-\varepsilon,\chi_k^{(2)}<\varepsilon\big)+{\mathbb{P}}\big(\chi_k^{(2)}>t-\varepsilon,\chi_k^{(1)}<\varepsilon\big).\end{align*}

By the previous part of the lemma we have that $n_k{\mathbb{P}}(\chi_k^{(1)}>\varepsilon,\chi_k^{(2)}>\varepsilon)$ converges to 0 as $k\rightarrow \infty$. Moreover, by independence of $\chi_k^{(1)}$ and $\chi_k^{(2)}$, for almost every $t>0$ we have that

\begin{equation*}\lim_{\varepsilon\rightarrow 0}\limsup_{k\rightarrow \infty}n_k\big|{\mathbb{P}}\big(\chi_k^{(1)}>t-\varepsilon\big){\mathbb{P}}\big(\chi_k^{(2)}<\varepsilon\big) -{\mathbb{P}}\big(\chi_k^{(1)}>t\big)\big|=0;\end{equation*}

therefore $\limsup_{k\rightarrow\infty}n_k({\mathbb{P}}(\Phi_n^{(2)}>t)-2{\mathbb{P}}(\Phi_n^{(1)}>t))\leq 0$, which completes the proof.□