1. Introduction and main result
Let
$\xi, \xi_1, \xi_2, \ldots$
be a sequence of independent and identically distributed (i.i.d.) random variables. Aguech et al. [Reference Aguech, Ben Hariz, El Machkouri and Faouzi1] introduced a new class of step-reinforced random walks, defined as follows: fix two parameters
$p, r\in [0,1]$
, set
$X_1=\xi_1$
, and for
$n\ge 2$
, define
$X_n$
recursively by
\begin{eqnarray}{X}_n \;:\!=\;\left\{\begin{array}{l@{\quad}l@{\quad}l} {X}_{U_n}, & \text{with probability} & rp,\\[5pt] -X_{U_n}, & \text{with probability} & (1-r)p, \\[5pt] \xi_{n}, & \text{with probability} & 1-p, \\[5pt] \end{array}\right. \end{eqnarray}
where
$\{U_n, n\ge 2\}$
is a sequence of independent random variables, independent of
$\{\xi_k\}$
, with each
$U_n$
uniformly distributed on
$ \{ 1,2, \ldots ,n-1 \}$
. The partial sum process
\begin{eqnarray}T_n=\sum_{k=1}^n X_k,\qquad n\ge 1, \end{eqnarray}
is called the unbalanced step-reinforced random walk.
A classical and widely studied instance of such a step-reinforced random walk is the elephant random walk (ERW), introduced by Schütz and Trimper [Reference Schütz and Trimper35]. The ERW has attracted considerable attention in recent years; see [Reference Baur and Bertoin4], [Reference Bercu5], [Reference Bertoin9], [Reference Coletti, Gava and Schütz15], [Reference Coletti, Gava and Schütz16], [Reference Dedecker, Fan, Hu and Merlevède18], [Reference Kubota and Takei29], [Reference Kürsten30], and [Reference Qin32]. As extensions of the ERW, the positively and negatively step-reinforced random walks were introduced by Simon [Reference Simon36] and Bertoin [Reference Bertoin10], respectively. They have been extensively investigated in [Reference Bertenghi and Rosales-Ortiz6], [Reference Bertoin7], [Reference Bertoin8], [Reference Bertoin10], [Reference Businger13], [Reference Hu24], [Reference Hu and Zhang25], and [Reference Qin33]. The unbalanced step-reinforced random walk serves as a unifying framework that includes all these models. Specifically, it reduces to the ERW when
$p=1$
and
$\mathbb{P}(\xi=1)=1-\mathbb{P}(\xi=-1)=s$
with
$s\in [0,1]$
, to the positively step-reinforced random walk when
$r=1$
, and to the negatively step-reinforced random walk when
$r=0$
. As in the cases of the ERW and the positively and negatively step-reinforced random walks, we refer to p as the memory parameter or the reinforcement parameter (see [Reference Bertenghi and Rosales-Ortiz6]).
Under the condition
$\mbox{Var}(\xi)\lt\infty$
, Aguech et al. [Reference Aguech, Ben Hariz, El Machkouri and Faouzi1] established the following limit theorems for the unbalanced step-reinforced random walk, where a phase transition occurs at
$a\;:\!=\;(2r-1)p=1/2$
.
Theorem 1.1. ([Reference Aguech, Ben Hariz, El Machkouri and Faouzi1]) Let
$\xi, \xi_1, \xi_2, \ldots$
be i.i.d. random variables with finite variance. Define
\begin{eqnarray} a\;:\!=\;(2r-1)p, \qquad \mu_1\;:\!=\;\frac{1-p}{1-a}\mathbb{E}(\xi), \qquad \sigma_1^2\;:\!=\;\mathbb{E}(\xi^2)-\mu_1^2. \end{eqnarray}
Then the following limit theorems hold.
-
(i) Subcritical regime (
$a\lt1/2$
): (1.4)
\begin{eqnarray} \frac{T_n-n\mu_1}{\sqrt{n}} \stackrel{\mathrm{d}}{\longrightarrow} N \bigg(0, \;\frac{\sigma_1^2}{1-2a}\bigg). \end{eqnarray}
-
(ii) Critical regime (
$a=1/2$
):
\begin{eqnarray*} \frac{T_n-n\mu_1}{\sqrt{n \log n}} \stackrel{\mathrm{d}}{\longrightarrow} N\big(0, \sigma_1^2\big). \end{eqnarray*}
-
(iii) Supercritical regime (
$a\gt1/2$
): where L is a square-integrable random variable.
\begin{eqnarray*} \frac{T_n-n\mu_1}{n^{a}} \longrightarrow L\;\;\textit{almost surely (a.s.)}, \end{eqnarray*}
Theorem 1.1 shows that, provided
$\xi$
has finite variance,
$T_n$
converges in distribution to the normal law in both the subcritical regime and the critical regimes. It is then natural to ask whether analogous convergence results remain valid when
$\xi$
belongs to the domain of attraction of the normal law but has infinite variance. In the subcritical regime, this question is answered affirmatively by the following theorem.
Theorem 1.2. Let
$\xi, \xi_1, \xi_2, \ldots$
be i.i.d. random variables with finite mean
$\mu\;:\!=\;\mathbb{E}(\xi)$
and infinite variance. Suppose further that
$\xi$
belongs to the domain of attraction of the normal law, i.e. there exist constants
$a_n\gt0$
such that
$a_n/\sqrt{n}\rightarrow \infty$
and
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n (\xi_k-\mu)\stackrel{\mathrm{d}}{\longrightarrow} N(0,1). \end{eqnarray}
Let a and
$\mu_1$
be defined as in (1.3). If
$a\lt1/2$
then
\begin{eqnarray} \frac{T_n-n\mu_1}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} N \bigg(0, \;\frac{1}{1-2a}\bigg). \end{eqnarray}
Remark 1.1. When
$\xi, \xi_1, \xi_2, \ldots$
are i.i.d. random variables with finite variance, the classical central limit theorem shows that (1.5) holds with
$a_n=\sqrt{n}\sigma$
, where
$\sigma^2=\mbox{Var}(\xi)$
. Moreover, (1.4) implies that
\begin{eqnarray*} \frac{T_n-n\mu_1}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} N \bigg(0, \;\frac{1}{1-2a}\cdot\frac{\sigma_1^2}{\sigma^2}\,\bigg), \end{eqnarray*}
where
$\sigma_1^2$
is defined as in (1.3). It is worth noting that, in contrast to (1.6), both cases yield normal limits, yet the limiting variances differ.
Furthermore, we consider the case where
$\xi$
belongs to the domain of attraction of some stable law with index
$\alpha\in (0,2)$
. In this case, we have (see [Reference Durrett20, Theorem 3.8.2] and [Reference Feller22, Chapter XVII, Section 5, Theorem 3])
\begin{eqnarray}\frac{1}{a_n}\sum_{k=1}^n (\xi_k-b_n)\stackrel{\mathrm{d}}{\longrightarrow} S, \end{eqnarray}
with
\begin{eqnarray}&& a_n\;:\!=\;\inf\{x\;:\; \mathbb{P}(|\xi|\gt x)\le n^{-1}\},\end{eqnarray}
\begin{eqnarray}&& b_n\;:\!=\;\left\{\begin{array}{l@{\quad}l} \mathbb{E}(\xi) & \mbox{if}\;\alpha\in (1,2), \\[5pt] \mathbb{E}(\xi I(|\xi|\le a_n)) & \mbox{if}\;\alpha = 1,\\[5pt] 0 & \mbox{if}\;\alpha \in (0, 1),\end{array}\right.\end{eqnarray}
and the characteristic function of S is given by
\begin{eqnarray}\mathbb{E}(\mathrm{e}^{itS})=\exp\{\mathrm{i}\gamma t-b|t|^{\alpha}(1+\mathrm{i}\kappa \;\mbox{sgn}(t) \omega(t,\alpha))\}, \end{eqnarray}
where
$\gamma\in \mathbb{R}, \alpha\in (0,2), b\gt0, \kappa\in [-1,1],\;\mbox{sgn}(t)=I(t\gt0)-I(t\lt0)$
is the sign function, and
\begin{eqnarray}\omega(t,\alpha)=\left\{\begin{array}{l@{\quad}l} \tan\! (\pi \alpha/{2}) & \mbox{if}\;\alpha \ne 1,\\[5pt] ({2}/{\pi}) \log |t| & \mbox{if}\;\alpha = 1.\end{array}\right. \end{eqnarray}
Theorem 1.3 establishes a non-Gaussian stable limit theorem for
$T_n$
in the case
$\kappa=0$
. To express the limit distribution, we introduce a special unbalanced step-reinforced random walk
$\{T_n^0, \;n\ge 1\}$
, defined via (1.1) and (1.2) with
$\xi\equiv 1$
and
$p=1$
. The moments of
$\{T_n^0, n\ge 1\}$
appear in the scaling constant of the limit distribution of
$T_n$
.
Theorem 1.3. Let
$\alpha \in (0,2), p\in (0,1),$
and
$ r\in [0,1]$
. Suppose that
$\xi, \xi_1, \xi_2, \ldots$
are i.i.d. random variables such that (1.7)–(1.10) hold with
$\kappa=0$
. If
$(2r-1) p\lt1/\alpha$
then
\begin{eqnarray} \frac{T_n-\delta nb_n}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} c(\alpha,p,r)^{1/\alpha} (S-\gamma)+\delta \gamma, \end{eqnarray}
where
$a=(2r-1)p, \;\delta=(1-p)/(1-a)$
, and
\begin{eqnarray*} c(\alpha,p,r)=\frac{1-p}{p}\sum_{k=1}^{\infty} \mathbb{E}\big(\big|T_k^0\big|^{\alpha}\big)\mathrm{B}(k, 1+1/p), \end{eqnarray*}
where B denotes the beta function.
Remark 1.2. Assume that
$\alpha\in (0, 2)$
. For the positively step-reinforced random walk in the subcritical regime (i.e.
$r=1$
and
$p\lt1/\alpha$
), Businger [Reference Businger13] obtained that if
$\mathbb{E}(\mathrm{e}^{\mathrm{i}t\xi})=\mathrm{e}^{-|t|^{\alpha}}$
then
$n^{-1/\alpha}T_n\stackrel{\mathrm{d}}{\longrightarrow} c(\alpha, p)^{1/\alpha} \xi$
, where
$ c(\alpha,p)=\int_0^{1-p} f(x)\,\mathrm{d}x $
with
\begin{eqnarray*} f(x)=\sum_{k=1}^{\infty}k^{\alpha} \bigg(1-\bigg(\frac{x}{1-p}\bigg)^p\bigg)^{k-1} \bigg(\frac{x}{1-p}\bigg)^p, \;\;\;\;x\in [0,1-p). \end{eqnarray*}
Note that in this case
$T_k^0=k$
, and a direct calculation shows that
\begin{eqnarray} c(\alpha,p,1)=\frac{1-p}{p}\sum_{k=1}^{\infty} k^{\alpha}\mathrm{B}(k, 1+1/p)=\int_0^{1-p} f(x)\,\mathrm{d}x. \end{eqnarray}
Thus, the constant
$c(\alpha, p)$
coincides with
$c(\alpha,p,1)$
. Bertoin [Reference Bertoin8] extended Businger’s result to the case where (1.7) holds with
$a_n=n^{-1/\alpha}$
and
$b_n=0$
. Bertoin [Reference Bertoin10] derived corresponding results for the negatively step-reinforced random walk, and the characteristic function of the limiting distribution is expressed in a complex form.
Remark 1.3. In Section 4, immediately before the proofs of the main theorems, we give a heuristic explanation of how the condition
$(2r-1)p \lt 1/\alpha$
arises.
In the special case where
$r=1$
and
$\mathbb{E}(\mathrm{e}^{\mathrm{i}t\xi})=\mathrm{e}^{-|t|^{\alpha}}$
for
$\alpha\in (0,2]$
, Businger [Reference Businger13] investigated the asymptotic behavior of
$T_n$
and observed a phase transition at the critical value
$p=1/\alpha$
.
-
(1) Subcritical regime (
$p\lt1/\alpha$
):
$n^{-1/\alpha}T_n\stackrel{\mathrm{d}}{\longrightarrow} (c(\alpha, p))^{1/\alpha} \xi$
, where
$c(\alpha, p)$
is defined as in Remark 1.2. -
(2) Critical regime (
$p=1/\alpha$
):
$(n\log n)^{-1/\alpha}T_n \stackrel{\mathrm{d}}{\longrightarrow} ((1-p)\Gamma(\alpha+1))^{1/\alpha^2}\xi$
. -
(3) Supercritical regime (
$p\gt1/\alpha$
):
$n^{-p} T_n\stackrel{\mathrm{P}}{\longrightarrow} Z$
, where Z is a random variable.
In view of the above result and Theorem 1.1, we conjecture that a phase transition occurs at
$(2r-1) p=1/\alpha$
under the conditions of Theorem 1.3. More precisely, in addition to the subcritical regime
$(2r-1)p \lt 1/\alpha$
established in Theorem 1.3, we expect the following behavior: at the critical value
$(2r-1) p=1/\alpha$
,
$T_n$
converges to a stable law, but with a normalization constant different from that in (1.12)—-possibly of the form
$a_nl(n)$
, where l(n) is slowly varying at infinity; when
$(2r-1)p\gt1/\alpha$
,
$T_n$
converges to a non-stable law. This conjecture remains open.
A key idea in the proofs of Theorems 1.2 and 1.3 is to represent
$T_n$
as a randomly weighted sum. More precisely, by establishing a connection with bond percolation on random recursive trees, we can express
$T_n$
as follows:
\begin{eqnarray}T_n = \sum_{k=1}^n W_{nk}\xi_k, \end{eqnarray}
where
$\{W_{nk}, n\ge 1, 1\le k\le n\}$
is independent of
$\{\xi_k, k\ge 1\}$
, as detailed in Section 4. This is a widely used technique in the study of step-reinforced random walks; see, e.g., [Reference Aguech and Qin2], [Reference Bertoin8], [Reference Bertoin10], [Reference Businger13], [Reference Hu24], and [Reference Hu and Zhang25].
In Subsection 2.2, we establish stable limit theorems for general randomly weighted sums, mainly using the Lindeberg replacement technique (see [Reference Feller22], Chapter VIII, Section 4). As an illustration of this method, we consider the special case of Theorem 1.3 in which the sequence
$\{\xi_k\}$
is symmetric. In this case, (1.7) and (1.10) hold with
$b_n=0$
and
$\gamma=\kappa=0$
. We then approximate
$\sum W_{nk} \xi_k /a_n$
by
$\sum W_{nk}S_k/ n^{1/\alpha},$
where
$S_1, S_2, \dots$
are i.i.d. copies of S and are independent of
$\{W_{nk}\}$
. This reduces the problem of determining the limiting distribution of
$\sum W_{nk} \xi_k /a_n$
to that of
$\sum W_{nk} S_k/n^{1/\alpha}$
. By applying the scaling property of S, we obtain
\begin{eqnarray*}\frac{1}{n^{1/\alpha}}\sum_{k=1}^n |W_{nk}| S_{k}\stackrel{\mathrm{d}}{=}\Bigg(\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha} \Bigg)^{1/\alpha} S.\end{eqnarray*}
Since S is symmetric, the weighted sums
$\sum_{k=1}^n W_{nk} S_{k}$
and
$\sum_{k=1}^n |W_{nk}| S_{k}$
have the same distribution. Combining this observation with the above approximation argument yields Theorem 1.3 in the symmetric case for
$\{\xi_k\}$
. For the general case, a slight modification of the above argument still works as long as
$\kappa=0$
; however, it fails when
$\kappa\neq 0$
. This is why the assumption
$\kappa=0$
is required in Theorem 1.3.
When
$r=1$
, the weights
$W_{nk}$
in (1.14) are non-negative and satisfy
$\sum_{k=1}^n W_{nk}=n$
for all
$n\ge 1$
. In this special case, the above method can be applied to arbitrary stable laws, leading to the following result.
Theorem 1.4. Assume that
$\alpha \in (0,2)$
,
$p\in (0,1)$
,
$r=1$
, and
$p\lt1/\alpha$
. Suppose
$\xi, \xi_1, \xi_2, \ldots$
are i.i.d. random variables such that (1.7)–(1.10) hold.
-
(a) If
$\alpha\in (0,1)\cup (1,2)$
then where
\begin{eqnarray*} \frac{T_n-nb_n}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} c(\alpha,p, 1)^{1/\alpha}(S-\gamma)+\gamma, \end{eqnarray*}
\begin{eqnarray*} c(\alpha, p, 1)=\frac{1-p}{p}\sum_{k=1}^{\infty} k^{\alpha}B(k, 1+1/p). \end{eqnarray*}
-
(b) If
$\alpha=1$
then where
\begin{eqnarray*} \frac{T_n- nb_n}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} S-2bc\kappa /\pi, \end{eqnarray*}
\begin{eqnarray*} c=\frac{1-p}{p}\sum_{k=1}^{\infty} k \log k B(k, 1+1/p). \end{eqnarray*}
Remark 1.4. Suppose
$\alpha$
, p, and r satisfy the assumptions in Theorem 1.4. Bertoin [Reference Bertoin8] considered a variant of the positively step-reinforced random walk and established stable limit theorems (Theorems 3 and 4 therein) when (1.7) holds with
$a_n = n^{1/\alpha}$
and
$b_n = 0$
. Theorem 1.4 extends these results.
The remainder of this paper is organized as follows. Subsection 2.1 reviews stable laws and their domains of attraction. Subsection 2.2 establishes three stable limit theorems (Theorems 2.1–2.3) for general randomly weighted sums. Their proofs are provided in Section 3 by our using the characteristic function method and the Lindeberg replacement technique. Finally, Section 4 presents the proofs of Theorems 1.2–1.4. Throughout this paper, C denotes a positive constant whose value may differ at each appearance.
2. Preliminaries
2.1. Stable laws and their domains of attraction
In this subsection, we review stable laws and their domains of attraction. The proofs of all the results in this subsection can be found in [Reference Feller22], [Reference Gnedenko and Kolmogorov23], and [Reference Ibragimov and Linnik26].
A distribution function F(x) and its characteristic function f(t) are called stable if, for every
$a_1\gt0$
and
$a_2\gt0$
, there exist
$a\gt0$
and b (depending on
$a_1$
and
$a_2$
) such that
\begin{align*}f(a_1t)f(a_2t)=\mathrm{e}^{\mathrm{i}bt}f(at).\end{align*}
Furthermore, f is stable if and only if it admits the representation
\begin{eqnarray}f(t)=\exp\{ \mathrm{i}\gamma t-b|t|^{\alpha}(1+\mathrm{i}\kappa \;\mbox{sgn}(t) \omega(t,\alpha))\}, \end{eqnarray}
where
$\gamma$
, b,
$\alpha$
, and
$\kappa$
are constants satisfying
$\gamma \in \mathbb{R}$
,
$b\ge 0$
,
$\alpha \in (0,2]$
, and
$\kappa\in [-1,1]$
, and
$\omega(t,\alpha)$
is defined as in (1.11). The case
$b=0$
corresponds to a degenerate law. When
$\alpha=2$
,
$f(t)=\mathrm{e}^{ \mathrm{i}\gamma t-bt^2}$
corresponds to the normal law
$N(\gamma, 2b)$
.
A fundamental problem in classical limit theory was to determine the possible limit laws of normalized sums
\begin{eqnarray*}Z_n=\frac{1}{a_n}\sum_{k=1}^n (\xi_k -b_n), \end{eqnarray*}
where
$\xi_1,\xi_2, \ldots $
are i.i.d. random variables and
$a_n\gt0$
, and
$b_n$
are suitable constants. A law F is stable if and only if it can be a limit law of
$Z_n$
.
For a given random variable
$\xi$
with law V, we say that
$\xi$
(or its law V) belongs to the domain of attraction of F if
$Z_n\stackrel{\mathrm{d}}{\longrightarrow} F$
for some constants
$a_n\gt0$
and
$b_n$
, where
$\xi_1,\xi_2, \ldots $
are i.i.d. copies of
$\xi$
. It should be noted that if two stable laws are both of the form (2.1) and differ only in the parameters
$\gamma$
and b, then their domains of attraction are identical. Lemmas 2.1 and 2.2 characterize the domains of attraction of the normal law and of non-Gaussian stable laws, respectively.
Lemma 2.1. A random variable
$\xi$
belongs to the domain of attraction of the normal law if and only if
$L(x)=\mathbb{E}(\xi^2I(|\xi|\le x))$
,
$x\ge 0$
, is slowly varying, i.e.
$L(tx)/L(x)\rightarrow 1$
as
$x\rightarrow \infty$
for any fixed
$t\gt0$
.
Assume that
$\xi$
belongs to the domain of attraction of the normal law. Then
$\mathbb{E}|\xi|^r\lt\infty$
for every
$0\lt r\lt2$
, and furthermore, there exists a slowly varying function l(x) such that
\begin{align*}\frac{1}{a_n}\sum_{k=1}^n (\xi_k-\mu)\stackrel{\mathrm{d}}{\longrightarrow} N(0,1),\end{align*}
where
$\xi_1,\xi_2, \ldots $
are i.i.d. copies of
$\xi$
,
$a_n=\sqrt{n}\,l(n)$
, and
$\mu=\mathbb{E}(\xi)$
. By the classical central limit theorem, every distribution with finite variance belongs to the normal domain of attraction; however, the normal domain of attraction also contains some distributions with infinite variance. In the finite-variance case, we may choose
$l(x)\equiv \sigma$
, where
$\sigma^2=\mbox{Var}(\xi)$
. In the infinite-variance case, l(x) is slowly varying, and
$l(x)\to\infty$
as
$x\to\infty$
.
Lemma 2.2. A random variable
$\xi$
belongs to the domain of attraction of a stable law with index
$\alpha\in (0,2)$
if and only if:
-
(1)
$\mathbb{P}(\xi\gt x)/\mathbb{P}(|\xi|\gt x)\rightarrow \theta\in [0,1]$
as
$x\rightarrow \infty$
; -
(2)
$\mathbb{P}(|\xi|\gt x)=x^{-\alpha} h(x)$
, where h(x) is slowly varying.
Assume
$\xi$
satisfies (1) and (2) above, and
$\xi_1,\xi_2, \ldots $
are i.i.d. copies of
$\xi$
. Then (1.7)–(1.10) hold with
$\kappa=2\theta-1$
. Moreover, we have
\begin{eqnarray}a_n=n^{1/\alpha}l(n), \end{eqnarray}
where l(x) is slowly varying.
2.2. Stable limit theorems for randomly weighted sums
Let
$\{\xi_k, k\ge 1\}$
be a sequence of i.i.d. random variables, and let
$\{W_{nk}, n\ge 1, 1\le k\le n\}$
be an array of random weights independent of
$\{\xi_k, k\ge 1 \}$
. Randomly weighted sums of the form
$\sum_{k=1}^n W_{nk}\xi_k$
play an important role in probability and statistics. Some examples of such sums are given below.
The bootstrap sample mean, introduced by Efron [Reference Efron21], is defined as
$\sum_{k=1}^n M_{nk}\xi_k ,$
where
$(M_{n1},\ldots, M_{nn})$
follows a multinomial distribution with n trials and success probability
$1/n$
per cell, and is independent of
$\{\xi_k \}$
.
A Bayesian analogue of the bootstrap was defined by Rubin [Reference Rubin34]. It is constructed by letting
$R_1, \ldots, R_{n-1}$
be i.i.d. uniform (0,1) random variables independent of
$\{\xi_k \}$
. If we denote their order statistics by
$R_{n1} \le \cdots \le R_{n,n-1} $
and set
$R_{n0}=0$
and
$R_{nn}=1$
, the gaps
$D_{nk}=R_{nk}-R_{n, k-1}$
for
$ 1\le k\le n$
form the weights, yielding the Bayesian bootstrap mean
$\sum_{k=1}^n D_{nk}\xi_k .$
Another example is the self-normalized sum studied by Breiman [Reference Breiman11], which is given by
$ \sum_{k=1}^n \xi_k Y_k/\sum_{k=1}^n Y_k$
, where we define
$0/0=0$
and
$\{Y_k\}$
is a sequence of i.i.d. non-negative, non-degenerate random variables independent of
$\{\xi_k \}$
.
A key feature in all the examples above is that the weights
$\{W_{nk}, 1\le k\le n\}$
are exchangeable for any fixed
$n\ge 1$
. However, applications often involve sums with non-exchangeable weights. The main objective of this subsection is to establish stable limit theorems for general randomly weighted sums. We begin with a simple Gaussian limit theorem obtained by applying the Lindeberg central limit theorem.
Theorem 2.1. Let
$\{\xi, \xi_k, k\ge 1\}$
be a sequence of i.i.d. random variables with
$\mathbb{E}(\xi)=0$
and
$\mathrm{Var}(\xi)=1$
. Suppose that
$\{W_{nk}, \;n\ge 1, 1\le k\le n\}$
is an array of random weights independent of
$\{\xi_k, k\ge 1 \}$
, satisfying the following conditions:
-
(A1)
$ \sum_{k=1}^n W_{nk}^2/n \stackrel{\mathrm{P}}{\longrightarrow} 1; $
-
(A2)
$\max\limits_{1\le k\le n}|W_{nk}|/\sqrt{n}\stackrel{\mathrm{P}}{\longrightarrow} 0$
.
Then
\begin{eqnarray} \frac{1}{\sqrt{n}}\sum_{k=1}^n W_{nk} \xi_k \stackrel{\mathrm{d}}{\longrightarrow} N(0,1). \end{eqnarray}
Remark 2.1. Suppose that the conditions of Theorem 2.1 hold, and further that the random weights
$W_{n1}, \ldots, W_{nn}$
are exchangeable for each
$n\ge 1$
. Under these conditions, Mason and Newton [Reference Mason and Newton31] established the following conditional central limit theorem:
\begin{eqnarray} \sup\nolimits_{x\in \mathbb{R}}\Bigg|\mathbb{P}\Bigg(\frac{1}{\sqrt{n}}\sum_{k=1}^n W_{nk} \xi_k \le x \Big| \xi_1, \xi_2, \ldots \Bigg)-\Phi(x)\Bigg|\rightarrow 0 \qquad \text{a.s.} \end{eqnarray}
In particular, (2.4) implies the unconditional convergence (2.3). Furthermore, as a consequence of a slight modification of Theorem 2.1 in [Reference Arenal-Gutiérrez and Matrán3], (2.3) and (2.4) are equivalent when
$\mbox{Var}(W_{n1})=O(1)$
and
$\sum_{k=1}^n W_{nk}=c_0n$
for some fixed constant
$c_0$
. However, for general weights
$\{W_{nk}\}$
, the conditional central limit theorem (2.4) may fail to hold.
Remark 2.2. Condition (A1) is necessary for the convergence
$ \sum_{k=1}^n W_{nk} \xi_k/\sqrt{n} \stackrel{\mathrm{d}}{\longrightarrow} N(0,1) $
to hold for any i.i.d. random variables
$\xi_1, \xi_2, \ldots$
with
$\mathbb{E}(\xi_1)=0$
and
$\mbox{Var}(\xi_1)=1$
. Indeed, let
$\xi_1, \xi_2, \ldots$
be i.i.d. N(0,1) random variables. If
$ \sum_{k=1}^n W_{nk} \xi_k /\sqrt{n}\stackrel{\mathrm{d}}{\longrightarrow} N(0,1) $
then
\begin{eqnarray*} \mathbb{E}\bigg(\!\exp\bigg\{\mathrm{i}t\frac{\sum_{k=1}^n W_{nk} \xi_k }{\sqrt{n}}\bigg\}\bigg)=\mathbb{E}\bigg(\!\exp\bigg\{-\frac{t^2}{2n}\sum_{k=1}^n W_{nk}^2\bigg\}\bigg)\rightarrow \mathrm{e}^{-t^2/2},\;\;t\in \mathbb{R}, \end{eqnarray*}
which implies condition (A1).
Remark 2.3. When the weights
$W_{nk}, \;n\ge 1, 1\le k\le n$
, are non-random, to study the limiting distribution of
$\sum_{k=1}^n W_{nk} \xi_k/\sqrt{n}$
, it is reasonable to assume that the sequence
$\{W_{nk} \xi_k /\sqrt{n}\}$
satisfies the infinite smallness condition. That is, for any
$\varepsilon\gt0$
,
\begin{eqnarray*} \sup\nolimits_{1\le k\le n} \mathbb{P} (|W_{nk}\xi_k /\sqrt{n}|\ge \varepsilon) =\mathbb{P} (|\xi|\ge \varepsilon/ M_n)\rightarrow 0, \end{eqnarray*}
where
$M_n=\max_{1\le k\le n}|W_{nk}|/\sqrt{n}$
. If the distribution function of
$\xi_i$
has unbounded support, then the above condition holds if and only if
$M_n\rightarrow 0$
. Therefore, condition (A2) can be regarded as a version of the infinite smallness condition. Below we give two examples where condition (A2) fails and the conclusion of Theorem 2.1 does not hold.
Example 2.1. For any
$n\ge 1$
, we set
$W_{n1}=\sqrt{n}$
and
$W_{nk}=0$
for
$2\le k\le n$
. Then condition (A1) holds, but
\begin{align*}\frac{1}{\sqrt{n}}\sum_{k=1}^n W_{nk} \xi_k=\xi_1.\end{align*}
Example 2.2. Let
$Y_1, Y_2, \ldots$
be i.i.d. non-negative random variables such that
$\mathbb{P}(Y_1\gt x)$
is slowly varying. By Theorem 1.2.1 in [Reference de Haan19], this is equivalent to
\begin{eqnarray*} \lim\nolimits_{x\rightarrow \infty}\frac{x^2\mathbb{P}(Y_1\gt x)}{\mathbb{E} Y_1^2I(Y_1\le x)} =\infty.\; \end{eqnarray*}
Define
$W_{nk}=\sqrt{n}Y_k/\sum_{i=1}^n Y_i$
for
$n\ge 1$
and
$1\le k\le n$
. Then, for each
$n\ge 1$
, the weights
$\{W_{nk}, 1\le k\le n\}$
are exchangeable. Moreover, condition (A1) holds (see Theorem 1.2 in [Reference Chistyakov and Götze14]):
\begin{align*}\frac{1}{n}\sum_{k=1}^n W_{nk}^2=\frac{\sum_{k=1}^n Y_k^2}{(\sum_{k=1}^n Y_k )^2}\stackrel{\mathrm{P}}{\rightarrow} 1.\end{align*}
If
$\xi, \xi_1, \xi_2, \ldots$
are i.i.d. random variables with
$\mathbb{E}|\xi|\lt\infty$
, then it follows from Corollary 1.11 in [Reference Kevei and Mason28] that
\begin{align*} \sum_{k=1}^n W_{nk} \xi_k/\sqrt{n}\stackrel{\mathrm{d}}{\rightarrow} \xi_1. \end{align*}
In this case, by Theorem 3.2 in [Reference Darling17], we have
$\max_{1\le k\le n}|W_{nk}|/\sqrt{n}\stackrel{\mathrm{P}}{\rightarrow} 1$
.
If
$\xi$
belongs to the domain of attraction of the normal law with
$\mathbb{E}(\xi)=0$
, then there exist constants
$a_n\gt0$
such that
\begin{eqnarray}\frac{1}{a_n}\sum_{k=1}^n \xi_k \stackrel{\mathrm{d}}{\longrightarrow} N(0,1). \end{eqnarray}
In view of Theorem 2.1, it is natural to investigate whether a corresponding central limit theorem holds for the randomly weighted sum:
\begin{eqnarray}\frac{1}{a_n}\sum_{k=1}^n W_{nk} \xi_k \stackrel{\mathrm{d}}{\longrightarrow} N(0,1). \end{eqnarray}
In this setting, conditions (A1) and (A2) are not sufficient to guarantee the convergence in (2.6).
Example 2.3. Let
$\xi$
be symmetric with
\begin{eqnarray*} \mathbb{P}(|\xi|\gt x)=\left\{ \begin{array}{l@{\quad}l} x^{-2}, & x\gt1,\\[5pt] 1, & x\le 1. \end{array} \right. \end{eqnarray*}
Then (2.5) holds with
$a_n=\sqrt{n\log n}$
(see Example 3.4.13 in [Reference Durrett20]). Let
$\{m(n), n=1,2,\ldots \}\subset \mathbb{N}$
be a sequence satisfying
$m(n)\rightarrow \infty$
and
$\log m(n)=o(\!\log n)$
. For any
$n\ge 1$
, define
\begin{eqnarray*} W_{nk}=\left\{ \begin{array}{l@{\quad}l} \sqrt{n/m(n)}, & 1\le k\le m(n), \\[5pt] 0, & m(n)\lt k\le n. \end{array} \right. \end{eqnarray*}
Then conditions (A1) and (A2) are satisfied, but
\begin{eqnarray*} \frac{\sqrt{\log n}}{\sqrt{\log m(n)}}\cdot\frac{1}{a_n}\sum_{k=1}^n W_{nk} \xi_k\stackrel{\mathrm{d}}{\longrightarrow} N(0,1), \end{eqnarray*}
which implies that (2.6) does not hold.
The following theorem establishes a general Gaussian limit theorem in the case where
$\xi$
belongs to the domain of attraction of the normal law.
Theorem 2.2. Let
$\{\xi, \xi_k, k\ge 1\}$
be a sequence of i.i.d. random variables with finite mean
$\mu=\mathbb{E}(\xi)$
. Suppose that
$\{W_{nk}, \;n\ge 1, 1\le k\le n\}$
is an array of random weights independent of
$\{\xi_k, k\ge 1 \}$
, satisfying the following conditions:
-
(A3)
$ \sum_{k=1}^n W_{nk}^2/n \stackrel{\mathrm{d}}{\longrightarrow} W, $
where W is some non-negative random variable; -
(A4)
$ \lim\limits_{c\rightarrow \infty} \sup\limits_n n^{-1}\sum_{k=1}^n \mathbb{E} (W_{nk}^{2} I(|W_{nk}|\gt c))=0. $
If
$\{a_n\}$
are positive constants such that
\begin{align*} \frac{1}{a_n}\sum_{k=1}^n (\xi_k-\mu) \stackrel{\mathrm{d}}{\longrightarrow} N(0,1), \end{align*}
then
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k -\mu) \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N, \end{eqnarray}
where N is a standard normal random variable independent of W.
Moreover, let
$\{c_n\}$
be a sequence of constants. If condition (A3) is replaced by the condition:
-
(A3ʹ) there exist random variables W and W’ such that
\begin{align*} \bigg(\frac{1}{n}\sum_{k=1}^n W_{nk}^2, \;\frac{1}{a_n}\sum_{k=1}^n (W_{nk}-c_n)\bigg) \stackrel{\mathrm{d}}{\longrightarrow} (W, W'); \end{align*}
then
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n (W_{nk} \xi_k -\mu c_n) \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N+\mu W', \end{eqnarray}
where N is a standard normal random variable independent of (W, W′).
Remark 2.4. Condition (A4) holds if there exists some
$\delta\gt0$
such that
\begin{eqnarray*} \sup_{n} \frac{1}{n}\sum_{k=1}^n \mathbb{E} \big(|W_{nk}|^{2+\delta}\big) \lt\infty. \end{eqnarray*}
More generally, condition (A4) is also satisfied if there exists a positive, non-decreasing function h(x) on
$\mathbb{R}^+$
such that
$h(x)\rightarrow \infty$
as
$x\rightarrow \infty$
and
$\sup_n \sum_{k=1}^n \mathbb{E}(W_{nk}^2h(|W_{nk}|))/n \lt\infty$
. Condition (A4) implies condition (A2), since for any
$\varepsilon\gt0$
,
\begin{eqnarray*} \mathbb{P}\Big(\max\limits_{1\le k\le n}|W_{nk}|/\sqrt{n}\ge \varepsilon \Big) \le \sum_{k=1}^n\mathbb{P} (|W_{nk}|\ge \sqrt{n}\varepsilon) \le\frac{1}{n\varepsilon^2}\sum_{k=1}^n \mathbb{E} \big(W_{nk}^{2} I(|W_{nk}|\gt\sqrt{n}\varepsilon)\big). \end{eqnarray*}
Similarly to Theorem 2.2, a non-Gaussian stable limit theorem can also be established for randomly weighted sums
$\sum_{k=1}^n W_{nk} \xi_k$
when
$\xi$
belongs to the domain of attraction of a stable law with index
$\alpha\in(0,2)$
.
Theorem 2.3. Let
$\xi, \xi_1, \xi_2, \ldots$
be i.i.d. random variables such that (1.7)–(1.10) hold with
$\kappa=0$
. Suppose that
$\{W_{nk}, \;n\ge 1, 1\le k\le n\}$
is an array of random weights independent of
$\{\xi_k, k\ge 1 \}$
, satisfying the following conditions:
-
(A5)
$ \sum_{k=1}^n |W_{nk}|^{\alpha}/n \stackrel{\mathrm{d}}{\longrightarrow} F, $
where F is a distribution function; -
(A6) there exists
$\beta\ge 1$
with
$\beta\gt\alpha$
such that
$ \lim\limits_{c\rightarrow \infty} \sup\limits_n \frac{1}{n}\sum_{k=1}^n \mathbb{E} (|W_{nk}|^{\beta} I(|W_{nk}|\gt c))=0. $
Then
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k-b_n)-\frac{\gamma}{n}\sum_{k=1}^n W_{nk} \stackrel{\mathrm{d}}{\longrightarrow} W^{1/\alpha} (S-\gamma), \end{eqnarray}
where W is a non-negative random variable with distribution function F and is independent of S.
Moreover, let
$\{c_n\}$
be a sequence of constants. If condition (A5) is replaced by the condition:
-
(A5ʹ) there exist random variables (W, W′, W′′), independent of S, such that
\begin{align*} \Bigg(\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha}, \;\frac{1}{n}\sum_{k=1}^n W_{nk}, \;\frac{b_n}{a_n}\sum_{k=1}^n (W_{nk}-c_n)\Bigg) \stackrel{\mathrm{d}}{\longrightarrow} (W, W', W''); \end{align*}
then
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n (W_{nk} \xi_k -b_nc_n) \stackrel{\mathrm{d}}{\longrightarrow} W^{1/\alpha} (S-\gamma)+\gamma W'+W''. \end{eqnarray}
Remark 2.5. Suppose further that the weights
$\{W_{nk}, n\ge 1, 1\le k\le n\}$
are non-negative and are independent of
$\{\xi_k, k\ge 1 \}$
. Theorem 2.3 can be extended to the domain of attraction of a general stable law. More precisely, if (1.7)–(1.10) hold then under conditions (A1) and (A6),
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k-b_n)-Y_{n,\alpha}\stackrel{\mathrm{d}}{\longrightarrow} W^{1/\alpha} (S-\gamma), \end{eqnarray}
where W is a non-negative random variable with distribution function F and is independent of S,
\begin{eqnarray*} Y_{n,\alpha}=\frac{\gamma}{n}\sum_{k=1}^n W_{nk}+\left\{ \begin{array}{l@{\quad}l} 0 & \mbox{if}\;\alpha\ne 1,\\ (2b\kappa/\pi)H_n & \mbox{if}\;\alpha= 1, \end{array} \right. \end{eqnarray*}
where
\begin{align*} H_n=-\frac{1}{n}\sum_{k=1}^n W_{nk} \log \Bigg(\frac{nW_{nk}}{\sum_{j=1}^n W_{nj}}\Bigg), \end{align*}
with the convention that
$0\log 0=0$
.
Theorems 2.1–2.3 have various applications, including the bootstrap sample mean
$\sum_{k=1}^n M_{nk}\xi_k$
, the Bayesian bootstrap mean
$\sum_{k=1}^n D_{nk}\xi_k$
, and the self-normalized sum
$ \sum_{k=1}^n \xi_k Y_k/\sum_{k=1}^n Y_k$
, all introduced at the beginning of this subsection. These applications lead to corresponding Gaussian and non-Gaussian stable limit theorems. However, since our focus is on unbalanced step-reinforced random walks, we do not pursue these applications further.
3. Proofs of Theorems 2.1–2.3
Proof of Theorem
2.1. For any sequence
$\{n_m, \;m=1,2,\ldots \}\subset \mathbb{N}$
, by conditions (A1) and (A2) there exists a subsequence
$\{n_{m_l}, \;l=1,2,\ldots \}$
such that, as
$l\rightarrow \infty$
,
\begin{eqnarray*} \frac{1}{n_{m_l}}\sum_{k=1}^{n_{m_l}} W_{n_{m_l},k}^2 \rightarrow 1\;\;\text{a.s.}\;\;\;\;\mbox{and}\;\;\;\;\max_{1\le k\le n_{m_l}}|W_{n_{m_l},k}|/\sqrt{n_{m_l}}\rightarrow 0 \;\;\text{a.s.} \end{eqnarray*}
Throughout this section, let
$\mathbb{P}_*$
and
$\mathbb{E}_*$
denote the conditional probability and expectation, respectively, given
$\{W_{nk}, \;n\ge 1, \;k=1,\ldots, n\}$
. Under
$\mathbb{P}_*$
, the array
$\{W_{n_{m_l},k}\xi_k /\sqrt{n_{m_l}},$
$k=1,\ldots, n_{m_l}, \;l=1,2,\ldots\}$
satisfies the conditions of the classical Lindeberg–Feller theorem (see Theorem 3.4.10 in [Reference Durrett20]) a.s. Hence,
\begin{eqnarray*} \sup\nolimits_{x\in \mathbb{R}}\Bigg| \mathbb{P}_* \Bigg(\frac{1}{\sqrt{n_{m_l}}}\sum_{k=1}^{n_{m_l}} W_{n_{m_l},k} \xi_k \le x \Bigg)- \Phi(x)\Bigg|\rightarrow 0\;\;\mbox{a.s. as}\;l\rightarrow \infty. \end{eqnarray*}
This implies that
\begin{eqnarray*} \sup\nolimits_{x\in \mathbb{R}}\Bigg| \mathbb{P} \Bigg(\frac{1}{\sqrt{n_{m_l}}}\sum_{k=1}^{n_{m_l}} W_{n_{m_l},k} \xi_k \le x \Bigg)-\Phi(x)\Bigg|\rightarrow 0\;\mbox{as}\;l\rightarrow \infty, \end{eqnarray*}
and the desired result follows.
Proof of Theorem 2.2. Define
\begin{eqnarray*} L(x)=\mathbb{E} ((\xi-\mu)^2I(|\xi|\le x)),\;\;x\in \mathbb{R}. \end{eqnarray*}
Then L(x) is a slowly varying function. Moreover (see, e.g., Section 5 in Chapter XVII of [Reference Feller22]), for any
$\varepsilon\gt0$
we have
\begin{eqnarray} nP(|\xi|\gt\varepsilon a_n)\rightarrow 0,\;\;\;\; \frac{n}{a_n}\mathbb{E}((\xi-\mu)I(|\xi|\le \varepsilon a_n))\rightarrow 0, \;\;\;\;\frac{nL(\varepsilon a_n) }{a_n^2} \rightarrow 1. \end{eqnarray}
Set
\begin{eqnarray*} \xi_{nk}=\frac{(\xi_k-\mu) I(|\xi_k |\le a_n)-\mathbb{E}((\xi-\mu)I(|\xi|\le a_n))}{a_n},\;\;\;\;\;\;n\ge 1, \;1\le k\le n, \end{eqnarray*}
and let
$Z_{n1}, \ldots, Z_{nn}$
be i.i.d.
$N(0,\sigma_{n1}^2)$
random variables independent of the weights
$\{W_{nk}\}$
, where
\begin{eqnarray} \sigma_{n1}^2=\mbox{Var}(\xi_{n1})=\frac{L(a_n)}{a_n^2}-\frac{(\mathbb{E}((\xi-\mu)I(|\xi|\le a_n)))^2}{a_n^2}\sim \frac{1}{n}. \end{eqnarray}
For any
$\varepsilon\in (0,1)$
, it follows from (3.1) that for sufficiently large n,
\begin{eqnarray} n\mathbb{E} \big(\xi_{n1}^2I(|\xi_{n1}|\gt \varepsilon)\big) &\le& n\mathbb{E} \big(\xi_{n1}^2I\big(|\xi_1|\gt \varepsilon a_n/2\big)\big)\nonumber\\[5pt] &\le& \frac{2n}{a_n^2}\mathbb{E} ((\xi-\mu)^2I(a_n\ge |\xi|\gt\varepsilon a_n/2))+\frac{2n}{a_n^2}(\mathbb{E}((\xi-\mu)I(|\xi|\le a_n)))^2\nonumber\\[5pt] &=&\frac{2nL(a_n)}{a_n^2}-\frac{2nL(\varepsilon a_n/2)}{a_n^2}+o(n^{-1})\rightarrow 0. \end{eqnarray}
Applying the elementary inequality
$|\mathrm{e}^{\mathrm{i}x}-1-\mathrm{i}x+x^2/2|\le \min\{x^2, |x|^3/6\}$
for
$x\in \mathbb{R}$
(see, for instance, Lemma 3.3.19 in [Reference Durrett20]) yields
\begin{eqnarray*} &&\Bigg|\mathbb{E}_* (\!\exp\{\mathrm{i}t W_{nk} \xi_{nk}\})-1+\frac{t^2}{2}W_{nk}^2 \sigma_{n1}^2\Bigg|\\[5pt] &\le & \mathbb{E}_* \Bigg|\exp\{\mathrm{i}t W_{nk} \xi_{nk}\}-1-\mathrm{i}tW_{nk}\xi_{nk}+\frac{t^2}{2}W_{nk}^2 \xi_{nk}^2\Bigg|\\[5pt] &\le& C\, \mathbb{E}_* \big(\big(W_{nk}^2 \xi_{nk}^2\big) \wedge |W_{nk}\xi_{nk}|^3\big), \end{eqnarray*}
where
$a\wedge b\;:\!=\;\min\{a,b\}$
for
$a,b\in \mathbb{R}$
. Similarly,
\begin{eqnarray*} \Bigg|\mathbb{E}_* (\!\exp \{\mathrm{i}t W_{nk} Z_{nk} \} )-1+\frac{t^2}{2}W_{nk}^2 \sigma_{n1}^2\Bigg| \le C\, \mathbb{E}_* \big(\big(W_{nk}^2 Z_{nk}^2\big) \wedge |W_{nk}Z_{nk}|^3\big). \end{eqnarray*}
Consequently,
\begin{eqnarray} &&\Bigg|\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} \xi_{nk}\Bigg\}\Bigg)-\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg)\Bigg|\nonumber\\[5pt] &\le&\mathbb{E}\Bigg|\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} \xi_{nk}\Bigg\}\Bigg)-\mathbb{E}_* \Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg)\Bigg|\nonumber\\[5pt] &=&\mathbb{E} \Bigg|\prod_{k=1}^n \mathbb{E}_* (\!\exp \{\mathrm{i}t W_{nk} \xi_{nk} \} )-\prod_{k=1}^n\mathbb{E}_* (\!\exp \{\mathrm{i}t W_{nk} Z_{nk} \} )\Bigg|\nonumber\\[5pt] &\le& \sum_{k=1}^n \mathbb{E} |\mathbb{E}_* (\!\exp \{\mathrm{i}t W_{nk} \xi_{nk} \})-\mathbb{E}_* (\!\exp \{\mathrm{i}t W_{nk} Z_{nk} \} )|\nonumber\\[5pt] &\le& C \sum_{k=1}^n \mathbb{E}\big(\big(W_{nk}^2 \xi_{nk}^2\big) \wedge |W_{nk}\xi_{nk}|^3\big)+ C \sum_{k=1}^n \mathbb{E}\big(\big(W_{nk}^2 Z_{nk}^2\big) \wedge |W_{nk}Z_{nk}|^3\big). \end{eqnarray}
We now estimate the terms on the right-hand side of (3.4). Using (3.2) and (3.3), we obtain
\begin{eqnarray*} &&\sum_{k=1}^n \mathbb{E} \big(W_{nk}^2 \xi_{nk}^2I(|W_{nk}\xi_{nk}|\gt\varepsilon)\big)\\[5pt] &\le & \sum_{k=1}^n \mathbb{E} \big(W_{nk}^2 \xi_{nk}^2 I(|W_{nk}|\gt c)\big)+\sum_{k=1}^n \mathbb{E} \big(W_{nk}^2 \xi_{nk}^2I(|\xi_{nk}|\gt\varepsilon/c)\big)\\[5pt] &\le & n \sigma_{n1}^2 \sup_m \frac{1}{m}\sum_{k=1}^m \mathbb{E} \big(W_{mk}^2 I(|W_{mk}|\gt c)\big) + \mathbb{E}\big(\xi_{n1}^2I(|\xi_{n1}|\gt\varepsilon/c)\big) \sum_{k=1}^n \mathbb{E} \big(W_{nk}^2\big) \rightarrow 0 \end{eqnarray*}
as
$n\rightarrow \infty$
and then
$c\rightarrow \infty$
, where we have used the independence between
$\{W_{nk}\}$
and
$\{\xi_k\}$
. Similarly,
\begin{eqnarray*} \sum_{k=1}^n \mathbb{E} (|W_{nk}\xi_{nk}|^3 I(|W_{nk}\xi_{nk}|\le \varepsilon)) \le \varepsilon \sum_{k=1}^n \mathbb{E} \big(W_{nk}^2 \xi_{nk}^2\big) =\varepsilon \sigma_{n1}^2\sum_{k=1}^n \mathbb{E} \big(W_{nk}^2\big) \rightarrow 0 \end{eqnarray*}
as
$n\rightarrow \infty$
and then
$\varepsilon\rightarrow 0$
, since
$\sup_n \sum_{k=1}^n \mathbb{E} (W_{nk}^2) /n\lt\infty$
by condition (A4). Hence,
\begin{eqnarray} \sum_{k=1}^n \mathbb{E} \big(\big(W_{nk}^2\xi_{nk}^{2}\big) \wedge |W_{nk}\xi_{nk}|^{3} \big) &\le & \sum_{k=1}^n \mathbb{E} \big(W_{nk}^2 \xi_{nk}^2I(|W_{nk}\xi_{nk}|\gt\varepsilon)\big)\nonumber\\[5pt] &&\;\;+ \sum_{k=1}^n \mathbb{E} \big(|W_{nk}\xi_{nk}|^3 I(|W_{nk}\xi_{nk}|\le \varepsilon)\big)\rightarrow 0\;\;\;\;\;\; \end{eqnarray}
as
$n\rightarrow \infty$
and then
$\varepsilon\rightarrow 0$
. A similar argument gives
\begin{eqnarray} \sum_{k=1}^n \mathbb{E} \big(\big(W_{nk}^2Z_{nk}^{2}\big) \wedge |W_{nk}Z_{nk}|^{3}\big)\rightarrow 0. \end{eqnarray}
On the other hand, by condition (A3),
\begin{eqnarray} \mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg) &=&\mathbb{E}\Bigg(\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg)\Bigg)\nonumber\\[5pt] &=&\mathbb{E}\Bigg(\!\exp\Bigg\{-\frac{t^2\sigma_{n1}^2}{2}\sum_{k=1}^n W_{nk}^2\Bigg\} \Bigg) \nonumber\\[5pt] &\rightarrow& \mathbb{E}\big(\mathrm{e}^{-t^2W/2}\big)=\mathbb{E}\big(\mathrm{e}^{it\sqrt{W}N}\big). \end{eqnarray}
Thus,
$ \sum_{k=1}^n W_{nk} Z_{nk} \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N$
. Combining this with (3.4)–(3.6) shows that
\begin{align*} \sum_{k=1}^n W_{nk} \xi_{nk}\stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N. \end{align*}
Now we relate the truncated sum to the original one. From (3.1) and condition (A3) we obtain
\begin{eqnarray} \!\!\!\! \Bigg|\frac{\mathbb{E}((\xi-\mu)I(|\xi|\le a_n))}{a_n}\sum_{k=1}^n W_{nk}\Bigg| \le \frac{n|\mathbb{E}((\xi-\mu)I(|\xi|\le a_n)) |}{a_n} \bigg(\frac{\sum_{k=1}^n W_{nk}^2}{n}\bigg)^{1/2} \stackrel{\mathrm{P}}{\longrightarrow} 0. \end{eqnarray}
Therefore,
\begin{align*} \frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k-\mu) I(|\xi_k |\le a_n) \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N. \end{align*}
Since, by (3.1),
\begin{eqnarray} &&\mathbb{P}\Bigg(\sum_{k=1}^n W_{nk} (\xi_k-\mu)\neq\sum_{k=1}^n W_{nk} (\xi_k-\mu) I(|\xi_k |\le a_n) \Bigg)\nonumber\\[5pt] &\le& \mathbb{P}\Bigg(\bigcup_{k=1}^n\{|\xi_k |\gt a_n\}\Bigg) \le\sum_{k=1}^n \mathbb{P}(|\xi_k |\gt a_n) =n\mathbb{P}(|\xi|\gt a_n) \rightarrow 0, \end{eqnarray}
we conclude that
\begin{align*} \frac{1}{a_n}\sum_{k=1}^n W_{nk}(\xi_k-\mu) \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W} N, \end{align*}
which is (2.7).
We now turn to the proof of (2.8). Notice that
$\exp\bigl(\mathrm{i}t\mu\sum_{k=1}^n (W_{nk}-c_n)/a_n\bigr)$
depends only on the weights. Hence, applying the same estimate as in (3.4) gives
\begin{eqnarray*} && \Bigg|\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n (W_{nk} \xi_{nk}+\mu (W_{nk}-c_n)/a_n)\Bigg\}\Bigg)\nonumber\\[5pt] &&\;\;\;\;\;\;\;\; -\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n (W_{nk} Z_{nk}+\mu (W_{nk}-c_n)/a_n)\Bigg\}\Bigg)\Bigg|\\[5pt] &\le&\mathbb{E}\Bigg|\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} \xi_{nk}\Bigg\}\Bigg)-\mathbb{E}_* \Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg)\Bigg|\rightarrow 0. \end{eqnarray*}
Proceeding as in (3.7) and using condition (A5ʹ), we have
\begin{eqnarray*} &&\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n (W_{nk} Z_{nk}+\mu (W_{nk}-c_n)/a_n)\Bigg\}\Bigg)\nonumber\\[5pt] &=&\mathbb{E}\Bigg(\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} Z_{nk}\Bigg\}\Bigg)\exp\Bigg\{\mathrm{i}t \mu \frac{\sum_{k=1}^n (W_{nk}-c_n)}{a_n}\Bigg\}\Bigg)\nonumber\\[5pt] &=&\mathbb{E}\Bigg(\!\exp\Bigg\{-\frac{t^2\sigma_{n1}^2}{2}\sum_{k=1}^n W_{nk}^2+\mathrm{i}t \mu \frac{\sum_{k=1}^n (W_{nk}-c_n)}{a_n}\Bigg\} \Bigg) \nonumber\\[5pt] &\rightarrow& \mathbb{E}\big(\mathrm{e}^{-t^2W/2+\mathrm{i}t\mu W'}\big)=\mathbb{E}\big(\mathrm{e}^{\mathrm{i}t(\sqrt{W}N+\mu W')}\big). \end{eqnarray*}
Therefore,
\begin{eqnarray*} \sum_{k=1}^n (W_{nk} \xi_{nk}+\mu (W_{nk}-c_n)/a_n) \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{W}N+\mu W'. \end{eqnarray*}
Proof of Theorem
2.3. Without loss of generality, we may assume that
$ \beta\in [1, 2]$
in condition (A6). For
$\alpha\in (0,2)$
, the characteristic exponent of
$\mathbb{E}(\mathrm{e}^{\mathrm{i}t(S-\gamma)})=\mathrm{e}^{-b|t|^{\alpha}}$
can be written in Lévy–Khinchin form (see, for instance, [Reference Breiman11, pp. 204–206]):
\begin{eqnarray*} -b|t|^{\alpha} =\mathrm{i}\tilde{\gamma} t+\int_{-\infty}^{\infty} f(t, x) \nu(\mathrm{d}x), \end{eqnarray*}
where
$\tilde{\gamma}\in \mathbb{R}$
,
$\nu$
is the Lévy measure with
$\nu(\{0\})=0$
,
$\nu(\mathrm{d}x)=\tilde{c} |x|^{-\alpha-1}\,\mathrm{d}x$
on
$\mathbb{R}-\{0\} (\tilde{c}\gt0)$
, and
\begin{eqnarray*} f(t, x)=\mathrm{e}^{\mathrm{i}tx}-1-\mathrm{i}txI(|x|\le 1). \end{eqnarray*}
Define
$\xi_{nk}=(\xi_k-b_n) /a_n-\gamma/n$
for
$1\le k \le n$
. Then
$\xi_{n1},\ldots, \xi_{nn}$
are i.i.d. for each
$n\ge 1$
, and
\begin{eqnarray*} \sum_{k=1}^n \xi_{nk}\stackrel{\mathrm{d}}{\longrightarrow} S-\gamma. \end{eqnarray*}
Moreover, we have (see Corollary 15.16 in [Reference Kallenberg27])
\begin{eqnarray} && n \mathbb{P}(\xi_{n1}\in \cdot)\stackrel{\mathrm{v}}{\longrightarrow} \nu, \;\;\;\;\; n\mathbb{E}(\xi_{n1}I(|\xi_{n1}|\le 1)) \rightarrow \tilde{\gamma}, \end{eqnarray}
\begin{eqnarray} && n \mathbb{E}(|\xi_{n1}|^{\beta}\wedge 1)\rightarrow \int (|x|^{\beta}\wedge 1)\,\mathrm{d}\nu(x)\lt\infty, \end{eqnarray}
where
$\stackrel{\mathrm{v}}{\rightarrow}$
denotes vague convergence.
Let
$S, S_1, S_2,\ldots$
be i.i.d. random variables independent of
$\{W_{nk}\}$
, and put
$S_{nk}\;:\!=\;n^{-1/\alpha} (S_k-\gamma) $
for
$1\le k\le n$
. From the characteristic function of S we then have
$ \sum_{k=1}^n S_{nk}\stackrel{\mathrm{d}}{=}S-\gamma $
and
\begin{eqnarray} \sum_{k=1}^n W_{nk}S_{nk} \stackrel{\mathrm{d}}{=}\Bigg(\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha} \Bigg)^{1/\alpha} (S-\gamma). \end{eqnarray}
Using the elementary inequality
\begin{eqnarray*} |f(t,x)|\le \frac{1}{2}|tx|^{\beta}I(|x|\le 1)+2I(|x|\gt1), \end{eqnarray*}
which holds for
$\beta\in [1,2]$
, we obtain that for any
$M\gt0$
,
\begin{eqnarray*} &&\Big|\mathbb{E}_* \big(\mathrm{e}^{\mathrm{i}tW_{nk}\xi_{nk}}\big)-1-\mathrm{i}t\tilde{\gamma} W_{nk}/n\Big|I(|W_{nk}|\gt M)\\[5pt] &=&\big|\mathrm{i}tW_{nk}(\mathbb{E}(\xi_{nk}I(|\xi_{nk}|\le 1))-\tilde{\gamma}/n)+\mathbb{E}_* (f(tW_{nk}, \xi_{nk}))\big|I(|W_{nk}|\gt M)\\[5pt] &\le& t\big|\mathbb{E}(\xi_{n1}I(|\xi_{n1}|\le 1))-\tilde{\gamma}/n)\big||W_{nk}|+ C\, \mathbb{E}(|\xi_{n1}|^{\beta} \wedge 1) |W_{nk}|^{\beta}I(|W_{nk}|\gt M). \end{eqnarray*}
Note that
$\sup_n \sum_{k=1}^n \mathbb{E}(|W_{nk}|)/n\lt\infty$
by condition (A6). It follows from (3.10), (3.11), and condition (A6) that
\begin{eqnarray*} \limsup_{M\rightarrow \infty}\limsup_{n\rightarrow \infty}\sum_{k=1}^n \mathbb{E}\Big(\Big|\mathbb{E}_* \big(\mathrm{e}^{\mathrm{i}tW_{nk}\xi_{nk}}\big)-1-\mathrm{i}t\tilde{\gamma} W_{nk}/n\Big|I(|W_{nk}|\gt M)\Big)=0. \end{eqnarray*}
Similarly,
\begin{eqnarray*} \limsup_{M\rightarrow \infty}\limsup_{n\rightarrow \infty}\sum_{k=1}^n \mathbb{E}\Big(\Big|\mathbb{E}_* \big(\mathrm{e}^{\mathrm{i}tW_{nk}S_{nk}}\big)-1-\mathrm{i}t\tilde{\gamma} W_{nk}/n\Big|I(|W_{nk}|\gt M)\Big)=0. \end{eqnarray*}
Consequently,
\begin{eqnarray} \limsup_{M\rightarrow \infty}\limsup_{n\rightarrow \infty}\sum_{k=1}^n \mathbb{E} \Big(\Big|\mathbb{E}_*\big(\mathrm{e}^{\mathrm{i}tW_{nk}\xi_{nk}}\big)-\mathbb{E}_* \big(\mathrm{e}^{\mathrm{i}tW_{nk}S_{nk}}\big)\Big|I(|W_{nk}|\gt M)\Big)=0. \end{eqnarray}
Since
$(\mathbb{E}\mathrm{e}^{\mathrm{i}t\xi_{n1}})^n\rightarrow \mathrm{e}^{-b|t|^{\alpha}}$
holds uniformly on any finite interval, applying Theorem 17.1 in [Reference Feller22] yields that
$n (\mathbb{E}\mathrm{e}^{\mathrm{i}t\xi_{n1}}-1)\rightarrow -b|t|^{\alpha}$
holds uniformly on any finite interval. Hence, for any fixed t, the convergence
$n (\mathbb{E}\mathrm{e}^{\mathrm{i}ta\xi_{n1}}-1)\rightarrow -b|ta|^{\alpha}$
is uniform in a on any finite interval. The same property holds for the sequence
$S_{n1}$
. Therefore, for any fixed
$M\gt0$
,
\begin{eqnarray*} &&\sum_{k=1}^n \mathbb{E} \Big(\Big|\mathbb{E}_*\big(\mathrm{e}^{\mathrm{i}tW_{nk}\xi_{nk}}\big)-\mathbb{E}_*\big(\mathrm{e}^{\mathrm{i}tW_{nk}S_{nk}}\big)\Big|I(|W_{nk}|\le M)\Big)\\[5pt] &\le& n \sup\nolimits_{|a|\le M} \Big|\mathbb{E}\big(\mathrm{e}^{\mathrm{i}ta\xi_{n1}}\big)-\mathbb{E}\big(\mathrm{e}^{\mathrm{i}taS_{n1}}\big)\Big|\\[5pt] &\le& \sup\nolimits_{|a|\le M} \Big|n\mathbb{E}\big(\mathrm{e}^{\mathrm{i}ta\xi_{n1}}-1\big)+b|ta|^{\alpha}\Big|+\sup_{|a|\le M} \Big|n\mathbb{E}\big(\mathrm{e}^{\mathrm{i}taS_{n1}}-1\big)+b|ta|^{\alpha}\Big|\rightarrow 0. \end{eqnarray*}
This together with (3.13) implies that
\begin{eqnarray} &&\Bigg|\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} \xi_{nk} \Bigg\}\Bigg)-\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} S_{nk} \Bigg\}\Bigg)\Bigg|\nonumber\\[5pt] &\le&\mathbb{E}\Bigg|\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk} \xi_{nk}\Bigg\}\Bigg)-\mathbb{E}_* \Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} S_{nk}\Bigg\}\Bigg)\Bigg|\nonumber\\[5pt] &\le& \sum_{k=1}^n \mathbb{E} \Big|\mathbb{E}_*\big(\mathrm{e}^{\mathrm{i}tW_{nk}\xi_{nk}}\big)-\mathbb{E}_*\big(\mathrm{e}^{\mathrm{i}tW_{nk}S_{nk}}\big)\Big|\rightarrow 0. \end{eqnarray}
Now (2.9) follows from (3.12), (3.14), and condition (A5). The proof of (2.10) follows by the same method as that of (2.8), so we omit the details.
Finally, we prove (2.11). To this end, we need the following lemma on stable random variables.
Lemma 3.1. Let
$S, S_1, S_2,\ldots$
be i.i.d. random variables such that S is stable with characteristic function (1.10). Let
$\{w_k\}$
be a sequence of real numbers. If
$\alpha\ne 1$
then
\begin{eqnarray*} \sum_{k=1}^n |w_k| (S_k-\gamma) \stackrel{\mathrm{d}}{=} \Bigg(\sum_{k=1}^n |w_k|^{\alpha}\Bigg)^{1/\alpha} (S-\gamma). \end{eqnarray*}
If
$\alpha=1$
then
\begin{eqnarray*} \sum_{k=1}^n |w_k|(S_k-\gamma)+\frac{2b\kappa}{\pi} \sum_{k=1}^n |w_k|\log \Bigg(\frac{ |w_k| }{\sum_{j=1}^n |w_j|}\Bigg)\stackrel{\mathrm{d}}{=} \sum_{k=1}^n |w_k| (S-\gamma). \end{eqnarray*}
Proof. The lemma follows directly from the characteristic function of S. The details are straightforward and therefore omitted.
Proof of (2.11). The proof is similar to that of Theorem 2.3, and we adopt the notation used there whenever no confusion arises.
For
$\alpha\in (0,2)$
, the characteristic exponent of S (see (1.10)) can be written in Lévy–Khinchin form (see, for instance, [Reference Breiman12, pp. 204--206]):
\begin{eqnarray*} \psi(t)=\mathrm{i}\gamma t-b|t|^{\alpha}(1+\mathrm{i}\kappa \;\mbox{sgn}(t) \omega(t,\alpha))=\mathrm{i}\tilde{\gamma} t+\int_{-\infty}^{\infty} f(t, x) \nu(\mathrm{d}x), \end{eqnarray*}
where
$\tilde{\gamma}\in \mathbb{R}$
,
$\nu$
is the Lévy measure with
$\nu(\{0\})=0$
,
$\nu(\mathrm{d}x)=\tilde{c}\theta x^{-\alpha-1}\mathrm{d}x$
on
$(0,\infty)$
and
$\nu(\mathrm{d}x)=\tilde{c}(1-\theta) |x|^{-\alpha-1}\,\mathrm{d}x$
on
$(-\infty, 0)$
,
$\tilde{c}\gt0$
,
$\theta=(\kappa+1)/2\in [0,1],$
and
\begin{eqnarray*} f(t, x)=\mathrm{e}^{\mathrm{i}tx}-1-\mathrm{i}txI(|x|\le 1). \end{eqnarray*}
Define
$\xi_{nk}=(\xi_k -b_n)/a_n$
for
$1\le k \le n$
. Then
$\xi_{n1},\ldots, \xi_{nn}$
are i.i.d. for each
$n\ge 1$
and (see Corollary 15.16 in [Reference Kallenberg27])
\begin{eqnarray*} && n \mathbb{P}(\xi_{n1}\in \cdot)\stackrel{\mathrm{v}}{\rightarrow} \nu, \;\;\;\;\; n\mathbb{E}(\xi_{n1}I(|\xi_{n1}|\le 1)) \rightarrow \tilde{\gamma}, \\[5pt] && n \mathbb{E}(|\xi_{n1}|^{\beta}\wedge 1)\rightarrow \int (|x|^{\beta}\wedge 1)\,\mathrm{d}\nu(x)\lt\infty, \end{eqnarray*}
where
$\stackrel{\mathrm{v}}{\rightarrow}$
denotes vague convergence. Let
$S, S_1, S_2,\ldots$
be i.i.d. random variables independent of
$\{W_{nk}\}$
. Define
$S_{nk}=n^{-1/\alpha}(S_k+b_n')$
for
$1\le k\le n$
with
\begin{eqnarray*} b_n'=\left\{ \begin{array}{l@{\quad}l} \gamma n^{1/\alpha-1}-\gamma & \mbox{if} \;\alpha\ne 1,\\[5pt] -(2b\kappa/\pi)\log n & \mbox{if} \;\alpha=1. \end{array} \right. \end{eqnarray*}
By Lemma 3.1, we have
$ \sum_{k=1}^n S_{nk}\stackrel{\mathrm{d}}{=}S $
and
\begin{eqnarray} \sum_{k=1}^n W_{nk}S_{nk}-Y_{n,\alpha}\stackrel{\mathrm{d}}{=}\Bigg(\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha} \Bigg)^{1/\alpha} (S-\gamma). \end{eqnarray}
Using arguments similar to those in the proof of (3.14), we obtain
\begin{eqnarray*} &&\Bigg|\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \Bigg(\sum_{k=1}^n W_{nk}\xi_{nk}-Y_{n,\alpha}\Bigg)\Bigg\}\Bigg)-\mathbb{E}\Bigg(\!\exp\Bigg\{\mathrm{i}t \Bigg(\sum_{k=1}^nW_{nk} S_{nk}-Y_{n,\alpha}\Bigg)\Bigg\}\Bigg)\Bigg|\\[5pt] &\le&\mathbb{E}\Bigg|\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^nW_{nk}\xi_{nk}\Bigg\}\Bigg)-\mathbb{E}_* \Bigg(\!\exp\Bigg\{\mathrm{i}t \sum_{k=1}^n W_{nk} S_{nk}\Bigg\}\Bigg)\Bigg|\\[5pt] &\le& \sum_{k=1}^n \mathbb{E} \Bigg|\mathbb{E}_*\Bigg(\!\exp\Bigg\{\mathrm{i}t W_{nk} \xi_{nk}\Bigg\}\Bigg)-\mathbb{E}_* \Bigg(\!\exp\Bigg\{\mathrm{i}t W_{nk} S_{nk}\Bigg\}\Bigg)\Bigg|\rightarrow 0, \end{eqnarray*}
and the desired result follows by (3.15) and condition (A5).
4. Proofs of main results
In this section, we prove Theorems 1.2–1.4 by applying the results on randomly weighted sums established in Subsection 2.2. To represent
$T_n$
explicitly as a randomly weighted sum of
$\{\xi_k, 1\le k\le n\}$
, we first introduce a refined definition of
$X_n$
.
Let
$\{\epsilon_n, n\ge 2\}$
and
$\{\eta_n, n\ge 2\}$
be independent sequences of i.i.d. Bernoulli random variables with parameters p and r, respectively. Assume further that these sequences are independent of
$\{\xi_n, n\ge 1\}$
and
$\{U_n, n\ge 2\}$
. Define
$X_1=\xi_1$
and, for
$n\ge 2$
,
\begin{eqnarray} {X}_n \;:\!=\;\left\{\begin{array}{c@{\quad}l} {X}_{U_n} & \text{if}\; \epsilon_n=1,\ \eta_n=1,\\[5pt] -X_{U_n} & \text{if}\; \epsilon_n=1,\ \eta_n=0, \\[5pt] \xi_{n} & \text{if}\; \epsilon_n=0.\end{array}\right.\end{eqnarray}
Now consider the tree
$\mathbb{T}_n$
with vertex set
$\{1,2, \ldots, n\}$
and edge set
$\{(U_k, k)\;:\; k=2,\ldots, n\}$
, which forms a random recursive tree of size n. Using
$\{\varepsilon_k, 2\le k\le n\}$
, we construct a Bernoulli bond percolation on
$\mathbb{T}_n$
with survival parameter
$1-p\in [0, 1]$
: for each
$2\le k\le n$
, the edge
$(U_k, k)$
is open if
$\epsilon_k=0$
and closed if
$\epsilon_k=1$
.
For each vertex k with
$\epsilon_k=0$
, we set
$W_{nk}=0$
. For vertices with
$\epsilon_k=1$
, we proceed as follows. Let such a vertex k be the root of its connected component in the percolation. Assign to each vertex of this component a value of
$+1$
or
$-1$
by traversing the tree in increasing order of labels: the root k receives 1; then, for any other vertex i (
$i\ne k$
) in the component, its value equals the value of its parent
$U_i$
if
$\eta_i=1$
, and the opposite value if
$\eta_i=0$
(note that
$U_i$
also belongs to the same component). The weight
$W_{nk}$
is then defined as the sum of all assigned values in this component.
With this construction we obtain the representation
\begin{eqnarray}T_n=\sum_{k=1}^n W_{nk}\xi_k, \end{eqnarray}
where the weights
$\{W_{nk}, \;n\ge 1, 1\le k\le n\}$
are independent of
$\{\xi_k, k\ge 1 \}$
. Moreover, in the proof of Theorems 1.2–1.4, we also use several other quantities related to the percolation. For any n, if j is the root of a component (i.e.
$\epsilon_j=1$
), let
$N_j(n)$
denote the size of that component; if
$\epsilon_j=0$
we set
$N_j(n)=0$
. Define
\begin{eqnarray*}\nu_k(n)\;:\!=\;\#\{1\le j\le n\;:\; N_j(n)=k\}\end{eqnarray*}
as the number of components of size k.
Before proving Theorems 1.2–1.4, we establish several preliminary lemmas. Recall that
$\{T_n^0, n\ge 1\}$
is a special unbalanced step-reinforced random walk defined before the statement of Theorem 1.3. Throughout, we set
$T_0^0=0$
by convention.
Lemma 4.1. Let
$n\ge 1$
be fixed and let
$m_1,\ldots, m_n$
be non-negative integers such that
$m_1+\cdots+m_n=n$
. Then, conditional on
$(N_1(n),\ldots, N_n(n))=(m_1, \ldots, m_n)$
, the weights
$\{W_{nk}, 1\le k\le n\}$
are independent, and for each
$k=1,\ldots, n$
we have
$W_{nk}\stackrel{\mathrm{d}}{=}T_{m_k}^0$
.
Proof. Analogously to (4.1), we give a refined definition of
$T_n^0$
. Set
$X_1^0\;:\!=\;1$
and, for
$n\ge 2$
,
\begin{eqnarray*} {X}_n^0\;:\!=\; \left\{ \begin{array}{l@{\quad}l} {X}^0_{U_n} & \text{if}\; \eta_n=1,\\[5pt] -X^0_{U_n} & \text{if}\; \eta_n=0. \end{array} \right. \end{eqnarray*}
Then
$ T_n^0\;:\!=\;\sum_{k=1}^n X^0_k $
is a version of the special unbalanced step-reinforced random walk.
With use of the constructions of
$W_{nk}$
and
$T_n^0$
, and the facts that
$\{\eta_n, n\ge 1\}$
are i.i.d. and independent of
$\{(U_n,\epsilon_n), n\ge 2\}$
, Lemma 4.1 follows directly from Lemma 3.3 in [Reference Bertoin10].
Lemma 4.2. For any
$r\in [0,1]$
and
$\beta\in (0,4]$
, there exists a constant
$C\gt0$
depending only on r such that
\begin{eqnarray*} \mathbb{E}\big(\big|T_n^0\big|^{\beta}\big)\le C a_r(n)^{\beta/2}, \end{eqnarray*}
where
\begin{eqnarray*} a_r(n)\;:\!=\;\left\{ \begin{array}{l@{\quad}l} n, & r\lt3/4,\\[5pt] n\log n, & r=3/4,\\[5pt] n^{4r-2}, & r\gt3/4. \end{array} \right. \end{eqnarray*}
Proof. In this proof, C denotes a positive constant depending only on r, which may vary from line to line. Write
$\gamma\;:\!=\;4r-2$
. It follows from the proof of Theorem 3.8 in [Reference Bercu5] that
\begin{eqnarray} &&\mathbb{E}\big(\big(T_{n+1}^0\big)^2\big)=\frac{n+\gamma}{n}\mathbb{E}\big(\big(T_n^0\big)^2\big)+1, \end{eqnarray}
\begin{eqnarray} &&\mathbb{E}\big(\big(T_{n+1}^0\big)^4\big)=1+\frac{6n+2\gamma}{n}\mathbb{E}\big(\big(T_n^0\big)^2\big)+ \frac{n+2\gamma}{n} \mathbb{E}\big(\big(T_n^0\big)^4\big). \end{eqnarray}
Using (4.3) and recalling
$T_1^0=1$
, we obtain
\begin{eqnarray} \mathbb{E}\big(\big(T_n^0\big)^2\big)=\frac{\Gamma(n+\gamma)}{\Gamma(n)}\sum_{k=1}^{n} \frac{\Gamma(k)}{\Gamma(k+\gamma)}, \end{eqnarray}
where
$\Gamma(\cdot)$
stands for the gamma function. Note that
$\Gamma(n+\gamma)/\Gamma(n)\sim n^{\gamma}$
as
$n\rightarrow \infty$
. If
$\gamma\le 1$
then
\begin{eqnarray*} \frac{\Gamma(n+\gamma)}{\Gamma(n)}\sum_{k=1}^{n} \frac{\Gamma(k)}{\Gamma(k+\gamma)} \sim n^{\gamma} \sum_{k=1}^n k^{-\gamma}\sim \left\{ \begin{array}{l@{\quad}l} n/(1-\gamma), & \gamma\lt1,\\[5pt] n\log n, & \gamma=1. \end{array} \right. \end{eqnarray*}
If
$\gamma\gt 1$
then
\begin{eqnarray*} c_0\;:\!=\;\sum_{k=1}^{\infty} \frac{\Gamma(k)}{\Gamma(k+\gamma)} \lt\infty, \end{eqnarray*}
and hence
$\mathbb{E}((T_n^0)^2)\sim c_0 n^{\gamma}$
. Moreover, by Lemma B.1 in [Reference Bercu5],
$c_0=1/((\gamma-1)\Gamma(\gamma))$
. Combining these facts yields
\begin{eqnarray*} \mathbb{E}\big(\big(T_n^0\big)^2\big) \le \left\{ \begin{array}{l@{\quad}l} Cn/(3-4r), & r\lt3/4,\\[5pt] Cn\log n, & r=3/4,\\[5pt] Cn^{4r-2}, & r\gt3/4. \end{array} \right. \end{eqnarray*}
Similarly, by applying (4.4), we get
\begin{eqnarray*} \mathbb{E}\big(\big(T_n^0\big)^4\big)=\frac{\Gamma(n+2\gamma)}{\Gamma(n)}\sum_{k=1}^{n} \frac{\Gamma(k)}{\Gamma(k+2\gamma)}d_{k-1} \le \left\{ \begin{array}{l@{\quad}l} Cn^2/(3-4r), & r\lt3/4,\\[5pt] Cn^2\log^2 n, & r=3/4,\\[5pt] C n^{8r-4}, & r\gt3/4, \end{array} \right. \end{eqnarray*}
where
$d_0=1$
and
\begin{eqnarray*} d_k=1+\frac{6k+2\gamma}{k}\mathbb{E}\big(\big(T_k^0\big)^2\big),\qquad k\ge 1. \end{eqnarray*}
Hence, for any
$\beta\in (0,4]$
,
\begin{eqnarray*} \mathbb{E}\big(\big|T_n^0\big|^{\beta}\big)\le \big(\mathbb{E}\big(\big(T_n^0\big)^4\big)\big)^{\beta/4}\le C\,a_r(n)^{\beta/2}. \end{eqnarray*}
This completes the proof of Lemma 4.2.
For two sequences of positive numbers
$(c_n)$
and
$(d_n)$
, we write
$c_n \asymp d_n$
if
$0\lt\liminf_{n\rightarrow\infty} c_n/d_n\le \limsup_{n\rightarrow\infty} c_n/d_n\lt\infty$
.
Lemma 4.3. Define
$ Z_{l}(n)=\sum_{k=1}^n k^l \nu_k(n). $
For any
$0\lt p\lt1$
and
$l\ge 0$
, we have
\begin{eqnarray} \mathbb{E}(Z_{l}(n)) \asymp b_{l}(n), \end{eqnarray}
where
\begin{align*} b_{l}(n)\;:\!=\;\left\{ \begin{array}{l@{\quad}l} n^{lp}, & lp\gt1,\\[5pt] n\log n, & lp=1,\\[5pt] n, & lp\lt1. \end{array} \right. \end{align*}
Furthermore, if
$lp\lt1$
then
\begin{eqnarray} \limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty} \frac{1}{n}\sum_{k=m}^n k^l \mathbb{E}(\nu_k(n)) =0. \end{eqnarray}
Proof. We prove the lemma in two main parts: first the moment estimate (4.6), and then the tail sum limit (4.7).
Part 1: Proof of (4.6).
We first show that there exists a constant
$C_l\gt0$
such that
$\mathbb{E}(Z_l(n))\le C_l b_l(n)$
for all
$n\in \mathbb{N}$
and all
$l\ge0$
. Throughout the proof of Lemma 4.3,
$C_l$
denotes a positive constant depending only on p and l, which may take a different value in each appearance.
The argument proceeds by induction on
$i\in\mathbb{N}$
with the following statement: for all l satisfying
$i-1\le l\lt i$
, the inequality
$\mathbb{E}(Z_l(n))\le C_l b_l(n)$
holds.
The statement is clear for
$i=1$
, since for any
$0\le l\lt1$
we have
\begin{align*}\mathbb{E}(Z_{l}(n))\le \mathbb{E}\Bigg(\sum_{k=1}^n k \nu_k(n)\Bigg)=n=b_l(n).\end{align*}
Assume the result holds for
$i=r-1$
. We proceed to the case
$i=r$
.
If
$\epsilon_{n+1}=1$
and
$U_{n+1}$
belongs to a component of size k at time n, then
$\nu_k(n+1)=$
$\nu_k(n)-1, \nu_{k+1}(n+1)=\nu_{k+1}(n)+1$
, and
$\nu_j(n+1)=\nu_j(n)$
for any
$j\not \in \{k, k+1\}$
. Hence, on this event,
$ Z_{l}(n+1)-Z_{l}(n)= (k+1)^l-k^l. $
Consequently,
\begin{eqnarray*} \mathbb{E}(Z_{l}(n+1)-Z_{l}(n)|\mathscr{F}_n, \epsilon_{n+1}=1) =\sum_{k=1}^{n} \frac{k\nu_{k}(n)}{n} ((k+1)^l-k^l), \end{eqnarray*}
where
$\mathscr{F}_n=\sigma(U_2,\ldots, U_n)$
. When
$\epsilon_{n+1}=0$
, we simply have
$Z_{l}(n+1)-Z_{l}(n)=1$
. Therefore,
\begin{eqnarray} \mathbb{E}(Z_{l}(n+1)-Z_{l}(n)|\mathscr{F}_n)=1-p+p\sum_{k=1}^{n} \frac{k\nu_{k}(n)}{n} ((k+1)^l-k^l). \end{eqnarray}
By Taylor’s theorem, for any
$k, l\ge 1$
,
\begin{eqnarray*} (k+1)^l-k^l-lk^{l-1}=\frac{l(l-1)}{2}\zeta_k^{l-2}\le l(l-1)2^{|l-2|-1} k^{l-2}, \end{eqnarray*}
where
$k\lt\zeta_k\lt k+1$
. Substituting this into (4.8) and taking expectations, we obtain
\begin{align*} \mathbb{E}(Z_{l}(n+1))-\mathbb{E}(Z_{l}(n)) &= 1-p+p\sum_{k=1}^{n} \frac{k \mathbb{E}(\nu_{k}(n))}{n} ((k+1)^l-k^l)\\[5pt] &\le 1-p+lpn^{-1}\mathbb{E}(Z_{l}(n))+C_ln^{-1}\mathbb{E}(Z_{l-1}(n)). \end{align*}
Using the induction hypothesis, we thus obtain
\begin{eqnarray} \mathbb{E}(Z_{l}(n+1)) \le \frac{n+lp}{n}\mathbb{E}(Z_{l}(n))+C_ln^{-1}b_{l-1}(n), \end{eqnarray}
where
\begin{eqnarray*} b_{l-1}(n)= \left\{ \begin{array}{c@{\quad}c} n^{(l-1)p}, & (l-1)p\gt1,\\[5pt] n\log n, & (l-1)p=1,\\[5pt] n, & (l-1)p\lt1. \end{array} \right. \end{eqnarray*}
We now solve (4.9) by distinguishing three cases according to the values of lp and
$(l-1)p$
.
Case 1:
$lp=1$
. Then
$b_{l-1}(n)=n$
and
$ \mathbb{E}(Z_{l}(n+1))\le (1+1/n)\mathbb{E}(Z_{l}(n))+C_l. $
Hence, for
$n\ge 2$
,
\begin{eqnarray*} \mathbb{E}(Z_{l}(n))\le C_ln\sum_{k=1}^{n} \frac{1}{k}\le C_ln\log n=C_lb_l(n). \end{eqnarray*}
Case 2:
$lp\ne 1$
and
$(l-1)p\lt1$
. Then
$b_{l-1}(n)=n$
and by (4·9),
\begin{eqnarray*} \mathbb{E}(Z_{l}(n+1))+\frac{C_l(n+1)}{lp-1} \le \frac{n+lp}{n}\bigg(\mathbb{E}(Z_{l}(n))+\frac{C_ln}{lp-1} \bigg). \end{eqnarray*}
This implies that
\begin{eqnarray*} \mathbb{E}(Z_{l}(n))+\frac{C_ln}{lp-1} \le \bigg(1+\frac{C_l}{lp-1}\bigg)a_{n}(l), \end{eqnarray*}
where
\begin{eqnarray*} a_n(l) \;:\!=\; \prod_{k=1}^{n-1} \frac{k+lp}{k} = \frac{\Gamma(n+lp)}{\Gamma(n)\Gamma(lp+1)}=\frac{n^{lp}}{\Gamma(lp+1)} (1+O(n^{-1})). \end{eqnarray*}
Consequently,
\begin{eqnarray*} \mathbb{E}(Z_{l}(n))\le C_ln^{(lp)\vee 1}=C_lb_l(n). \end{eqnarray*}
Case 3:
$(l-1)p\ge 1$
. Then for large n,
\begin{eqnarray*} b_{l-1}(n+1)-b_{l-1}(n)\le (l-1/2)pn^{-1}b_{l-1}(n). \end{eqnarray*}
This together with (4.9) yields
\begin{eqnarray*} \mathbb{E}(Z_{l}(n+1))+\frac{2C_lb_{l-1}(n+1)}{p} \le \frac{n+lp}{n}\bigg(\mathbb{E}(Z_{l}(n))+\frac{2C_lb_{l-1}(n)}{p}\bigg). \end{eqnarray*}
Hence,
\begin{eqnarray*} \mathbb{E}(Z_{l}(n))\le C_ln^{lp}=C_lb_l(n). \end{eqnarray*}
The induction is thus complete, establishing
$\mathbb{E}(Z_{l}(n)) \le C_l b_{l}(n)$
for all
$l\ge 0$
.
Similarly, we can establish the lower bound
$\mathbb{E}(Z_{l}(n)) \ge C_l b_{l}(n)$
for
$l\ge 0$
. Hence (4.6) holds.
Part 2: Proof of (4.7).
The proof proceeds in two steps: first for
$0\le l\lt1$
, and then for general l by induction on the integer part of l.
Recall (see [Reference Simon36]) that for any fixed
$k\ge 1$
,
\begin{eqnarray*} \lim_{n\rightarrow \infty}\frac{\mathbb{E}(\nu_k(n))}{n}=\frac{1-p}{p} \mathrm{B}(k, 1+1/p). \end{eqnarray*}
Hence, for any
$0\le l\lt1$
and
$m\in \mathbb{N}$
, we get, as
$ n\rightarrow \infty$
,
\begin{eqnarray*} \frac{1}{n} \sum_{k=m}^n k^l \mathbb{E}(\nu_k(n)) &\le& \frac{1}{n} \sum_{k=m}^n k \mathbb{E}(\nu_k(n)) = 1-\frac{1}{n} \sum_{k=1}^{m-1} k \mathbb{E}(\nu_k(n))\\[5pt] &\rightarrow& 1-\frac{1-p}{p}\sum_{k=1}^{m-1} k\mathrm{B}(k, 1+1/p), \end{eqnarray*}
where we have used the fact
$\sum_{k=1}^n k\nu_k(n)=n$
. Because of the identity (see (8) in [Reference Bertoin10])
\begin{eqnarray*} \sum_{k=1}^{\infty}\frac{1-p}{p} k\mathrm{B}(k, 1+1/p)=1, \end{eqnarray*}
we obtain
\begin{eqnarray*} \limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty}\frac{1}{n} \sum_{k=m}^n k^l \mathbb{E}(\nu_k(n)) \le \limsup_{m\rightarrow \infty} \frac{1-p}{p}\sum_{k=m}^{\infty} k\mathrm{B}(k, 1+1/p)=0, \end{eqnarray*}
which is (4.7) for
$0\le l\lt1$
.
Now suppose (4.7) holds for all l with
$r-1\le l\lt r$
and
$lp\lt1$
, where
$r\in \mathbb{N}$
. We prove it for l satisfying
$r\le l\lt r+1$
and
$lp\lt1$
. Fix such an l. By Stirling’s formula,
$\mathrm{B}(m, 1+1/p)\asymp m^{-1-1/p}$
. Consequently,
\begin{eqnarray*} \limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty}\frac{m^l \mathbb{E}(\nu_m(n))}{n}= \limsup_{m\rightarrow \infty}\frac{1-p}{p} m^l\mathrm{B}(m, 1+1/p)=0. \end{eqnarray*}
Therefore, for any
$\varepsilon\gt0$
, there exists
$m=m(\varepsilon)$
such that
\begin{eqnarray*} \limsup_{n\rightarrow \infty}\frac{m^l \mathbb{E}(\nu_m(n))}{n}\lt\varepsilon, \qquad\limsup_{n\rightarrow \infty} \frac{1}{n}\sum_{k=m}^n k^{l-1} \mathbb{E}(\nu_k(n)) \lt\varepsilon. \end{eqnarray*}
Using arguments similar to those in the proof of (4.6), we obtain, for sufficiently large n,
\begin{align*} \sum_{k=m}^{n+1} k^{l} \mathbb{E}(\nu_k(n+1))&\le \frac{n+lp}{n}\sum_{k=m}^n k^{l} \mathbb{E}(\nu_k(n))+\frac{m^l \mathbb{E}(\nu_{m}(n+1))}{n}+C_ln^{-1}\sum_{k=m}^n k^{l-1} \mathbb{E}(\nu_k(n))\\[5pt] &\le \frac{n+lp}{n}\sum_{k=m}^n k^{l} \mathbb{E}(\nu_k(n))+C_l\varepsilon, \end{align*}
and, consequently,
\begin{eqnarray*} \frac{1}{n}\sum_{k=m}^n k^l \mathbb{E}(\nu_k(n))\le C_l \varepsilon. \end{eqnarray*}
Hence, (4.7) holds for
$r\le l\lt r+1$
with
$lp\lt1$
. This completes the induction step and the proof of (4.7).
Lemma 4.4. Let
$\alpha \in (0,2], r\in [0,1]$
, and
$p\in (0,1)$
be such that
$(2r-1)\alpha p\lt1$
. Assume
$g\;:\; \{0, 1, \ldots\}\rightarrow [0,\infty)$
satisfies
$g(0)=0$
and
$g(x)=O( x^{\alpha})$
as
$x\rightarrow \infty$
. Then
\begin{eqnarray} \frac{\sum_{k=1}^n g(W_{nk})}{n}\stackrel{\mathrm{P}}{\longrightarrow} c_g, \end{eqnarray}
where
\begin{eqnarray*} c_g=\frac{1-p}{p}\sum_{k=1}^{\infty} g(k)\mathrm{B}(k, 1+1/p). \end{eqnarray*}
Proof. Let
$\{T_{nk}^0, n,k\ge 1\}$
be a sequence of independent random variables, independent of
$\{\nu_{k}(n), n,k\ge 1\}$
, such that
$T_{nk}^0\stackrel{d}{=} T_n^0$
for all
$n,k\ge 1$
. By Lemma 4.1,
\begin{eqnarray} \frac{1}{n}\sum_{k=1}^n g(|W_{nk}|) \stackrel{\mathrm{d}}{=}\frac{1}{n}\sum_{k=1}^n \sum_{i=1}^{\nu_k(n)} g\big(\big|T_{ki}^0\big|\big). \end{eqnarray}
Recall that (see Lemma 3.1 in [Reference Bertoin10])
\begin{eqnarray} \frac{\nu_k(n)}{n} \stackrel{\mathrm{P}}{\longrightarrow} \frac{1-p}{p} \mathrm{B} (k, 1+1/p). \end{eqnarray}
Consequently, for any fixed
$k\ge 1$
, we have
\begin{eqnarray} \frac{1}{n}\sum_{i=1}^{\nu_k(n)} g\big(\big|T_{ki}^0\big|\big)= \frac{\nu_k(n)}{n} \frac{1}{\nu_k(n)}\sum_{i=1}^{\nu_k(n)} g\big(\big|T_{ki}^0\big|\big) \stackrel{\mathrm{P}}{\longrightarrow} \frac{1-p}{p} \mathbb{E}\big(g\big(\big|T_k^0\big|\big)\big)\mathrm{B}(k, 1+1/p). \end{eqnarray}
Application of Stirling’s formula gives
$\mathrm{B}(k, 1+1/p)\sim C_p k^{-1-1/p}$
as
$k\rightarrow \infty$
, where
$C_p\gt0$
is a constant depending only on p. Hence, if
$(2r-1)\alpha p\lt1$
then it follows from Lemma 4.2 that
\begin{eqnarray} &&\limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty}\sum_{k=m+1}^n \mathbb{E}\big(g\big(\big|T_k^0\big|\big)\big)\mathrm{B}(k,1+1/p)\nonumber\\[5pt] &\le& C\limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty}\sum_{k=m+1}^n (a_r(k))^{\alpha/2}k^{-1-1/p}=0. \end{eqnarray}
Now let
$\varepsilon\gt0$
. By (4.7) in Lemma 4.3,
\begin{eqnarray*} &&\limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty}\mathbb{P}\Bigg(\frac{1}{n}\sum_{k=m+1}^n \sum_{i=1}^{\nu_k(n)} g\big(\big|T_{ki}^0\big|\big)\gt\varepsilon \Bigg)\\[5pt] &\le&\limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty} \frac{1}{\varepsilon n}\sum_{k=m+1}^n \mathbb{E} \Bigg( \sum_{i=1}^{\nu_k(n)} g\big(\big|T_{ki}^0\big|\big)\Bigg)\\[5pt] &\le& \limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty} \frac{1}{\varepsilon n}\sum_{k=m+1}^n \mathbb{E}\big(g\big(\big|T_k^0\big|\big)\big)\mathbb{E} (\nu_k(n))\\[5pt] &\le& C\limsup_{m\rightarrow \infty}\limsup_{n\rightarrow \infty} \frac{1}{\varepsilon n}\sum_{k=m+1}^n (a_r(k))^{\alpha/2}\mathbb{E}(\nu_k(n))=0. \end{eqnarray*}
Combining this with (4.11), (4.13), and (4.14) yields (4.10).
We now turn to the proofs of Theorems 1.2–1.4, starting with Theorem 1.3. Before the formal proof, we first use the above lemmas to give a heuristic explanation of how the condition
$(2r-1)p\lt1/\alpha$
in Theorem 1.3 arises.
Recall from (4.2) that
$T_n=\sum_{k=1}^n W_{nk}\xi_k$
. Consider the special case where
$\xi_k$
are i.i.d. with characteristic function
$\mathbb{E}(\mathrm{e}^{\mathrm{i}t\xi_1})=\mathrm{e}^{-|t|^{\alpha}}$
, so that
$n^{-1/\alpha}\sum_{k=1}^n \xi_k \stackrel{\mathrm{d}}{=} \xi_1.$
The conclusion of Theorem 1.3 is
$T_n/n^{1/\alpha}\stackrel{\mathrm{d}}{\longrightarrow} \tilde{c}\,\xi_1$
for some constant
$\tilde{c}$
. In this case, we have
\begin{align*}\mathbb{E}\Bigg(\!\exp\Bigg\{-\frac{|t|^{\alpha}}{n}\sum_{k=1}^n |W_{nk}|^{\alpha}\Bigg\}\Bigg)=\mathbb{E}\big(\mathrm{e}^{\mathrm{i}t T_n/n^{1/\alpha}}\big)\to \mathrm{e}^{-(\tilde{c}|t|)^{\alpha}},\end{align*}
which implies
\begin{align*}\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha} \stackrel{\mathrm{P}}{\longrightarrow} (\tilde{c})^{\alpha}.\end{align*}
With use of Lemmas 4.1 and 4.2 together with (4.11) and (4.12), a heuristic argument leads to
\begin{align*}\frac{1}{n}\sum_{k=1}^n |W_{nk}|^{\alpha} &\stackrel{\mathrm{d}}{=} \frac{1}{n}\sum_{k=1}^n \sum_{i=1}^{\nu_k(n)} |T_{ki}^0|^{\alpha}\approx \sum_{k=1}^n \mathbb{E}\big(|T_k^0|^{\alpha}\big) \, \mathrm{B}(k, 1+1/p)\\&= O(1) \sum_{k=1}^n a_r(k)^{\alpha/2} \, \mathrm{B}(k, 1+1/p),\end{align*}
where
$a_r(k)$
is defined as in Lemma 4.2. By Stirling’s formula,
$\mathrm{B}(k, 1+1/p) \asymp k^{-1-1/p}$
. Thus, the general term of the series satisfies
\begin{align*}a_r(k)^{\alpha/2} \, \mathrm{B}(k, 1+1/p)\asymp\left\{\begin{array}{l@{\quad}l} k^{\alpha/2 - 1 - 1/p}, & r\lt3/4,\\[5pt] k^{\alpha/2 - 1 - 1/p} (\!\log k)^{\alpha/2}, & r=3/4,\\[5pt] k^{\alpha(2r-1) - 1 - 1/p}, & r\gt3/4.\end{array}\right.\end{align*}
For the series to converge to a finite limit, the exponent of k must be less than
$-1$
. When
$r\gt3/4$
, this requires
$\alpha(2r-1) - 1 - 1/p \lt -1$
, i.e.
$(2r-1)p \lt 1/\alpha$
. When
$r\le 3/4$
, one can easily verify that
$(2r-1)p \lt 1/\alpha$
holds automatically, and the series converges as well. This explains why the assumption
$(2r-1)p\lt1/\alpha$
is needed in Theorem 1.3.
Proof of Theorem
1.3. Recall that
$a_n=n^{1/\alpha}l(n)$
(see (2.2)), where l(x) is slowly varying at infinity. We first show a simple fact:
\begin{eqnarray} \sqrt{n}\,b_n/a_n\rightarrow 0. \end{eqnarray}
The case
$\alpha\ne 1$
is obvious; we therefore focus on
$\alpha=1$
. Using (5.22) in [Reference Feller22, Chapter XVII, Section 5], we have that for any
$\beta_0\in (1,3/2)$
,
\begin{eqnarray*} |b_n| \le \mathbb{E}(|\xi|I(|\xi|\le a_n)) \le 1+\mathbb{E}(|\xi|^{\beta_0} I(|\xi|\le a_n)) \le Ca_n^{\beta_0-1}; \end{eqnarray*}
hence,
$\sqrt{n}\,|b_n|/a_n\le C\sqrt{n}\,a_n^{\beta_0-2} =Cn^{\beta_0-3/2}l(n)^{\beta_0-2}\rightarrow 0$
. Thus (4.15) holds.
Assume that
$(2r-1) p\lt1/\alpha$
. Then we can choose
$\beta$
such that
$\max\{\alpha, 1\}\lt\beta\lt4$
,
$(2r-1)\beta p\lt1$
, and
$\beta p\lt2$
. Using Lemma 4.3 and arguments similar to those in the proof of Lemma 4.4, we obtain
\begin{align} \sup \limits_n \frac{1}{n}\sum_{k=1}^n \mathbb{E} (|W_{nk}|^{\beta})&= \sup \limits_n \frac{1}{n}\sum_{k=1}^n \mathbb{E}\big(\big|T_k^0\big|^{\beta}\big)\mathbb{E} (\nu_k(n))\nonumber\\[5pt] &\le C\sup \limits_n \frac{1}{n}\sum_{k=1}^n (a_r(k))^{\beta/2}\mathbb{E} (\nu_k(n)) \lt\infty. \end{align}
Hence, condition (A6) holds.
By Lemma 4.4, we have
\begin{eqnarray} \frac{1}{n} \sum_{k=1}^n |W_{nk}|^{\alpha} \stackrel{\mathrm{P}}{\longrightarrow} c(\alpha, p, r). \end{eqnarray}
Applying Theorem 1.1(i) with
$\xi\equiv 1$
yields
\begin{eqnarray*} \frac{\sum_{k=1}^n W_{nk}-n\delta}{\sqrt{n}}\stackrel{\mathrm{d}}{\longrightarrow} N\bigg(0,\;\frac{1-\delta^2}{1-2a}\bigg), \end{eqnarray*}
where
$a=(2r-1)p$
and
$\delta=(1-p)/(1-a)$
. This implies
\begin{eqnarray} \frac{1}{n}\sum_{k=1}^n W_{nk}\stackrel{\mathrm{P}}{\longrightarrow} \delta, \end{eqnarray}
and together with (4.15) gives
\begin{eqnarray} \frac{b_n}{a_n}\sum_{k=1}^n (W_{nk}-\delta)\stackrel{\mathrm{P}}{\longrightarrow} 0. \end{eqnarray}
Combining (4.17), (4.18), and (4.19) and applying Theorem 2.3 yields (1.12), which completes the proof of Theorem 1.3.
Proof of Theorem 1.2. Applying Theorem 2.2 and arguments similar to those in the proof of Theorem 1.3 (details omitted) yields
\begin{eqnarray*} \frac{T_n-n\mu_1}{a_n} \stackrel{\mathrm{d}}{\longrightarrow} \sqrt{c(2,p,r)}\, N, \end{eqnarray*}
where N is a standard normal random variable and
\begin{eqnarray*} c(2,p,r)=\frac{1-p}{p}\sum_{n=1}^{\infty} \mathbb{E}\big(\big|T_n^0\big|^{2}\big)\mathrm{B}(n, 1+1/p). \end{eqnarray*}
Assume that
$a=(2r-1)p\lt1/2$
. To prove Theorem 1.2 it suffices to show that
\begin{eqnarray} c(2,p,r)=\frac{1}{1-2a}. \end{eqnarray}
From (4.5) we obtain
\begin{eqnarray*} c(2,p,r)&=&\frac{1-p}{p}\sum_{n=1}^{\infty} \frac{\Gamma(n+4r-2)\Gamma(1+1/p)}{\Gamma(n+1+1/p)}\sum_{k=1}^{n} \frac{\Gamma(k)}{\Gamma(k+4r-2)}\\[5pt] &=& \frac{1-p}{p}\sum_{k=1}^{\infty}\frac{\Gamma(k)\Gamma(1+1/p)}{\Gamma(k+4r-2)} \sum_{n=k}^{\infty} \frac{\Gamma(n+4r-2)}{\Gamma(n+1+1/p)}. \end{eqnarray*}
Note that for any
$k\in \mathbb{N}$
and any positive numbers a, b with
$a\lt b$
, the following telescoping identity holds:
\begin{eqnarray*} \sum_{n=k}^{\infty} \frac{\Gamma(n+a)}{\Gamma(n+b+1)} =\frac{1}{b-a}\sum_{n=k}^{\infty} \Big(\frac{\Gamma(n+a)}{\Gamma(n+b)}-\frac{\Gamma(n+a+1)}{\Gamma(n+b+1)}\Big) =\frac{\Gamma(k+a)}{\Gamma(k+b)(b-a)}. \end{eqnarray*}
Applying it twice yields
\begin{eqnarray*} c(2,p,r) &=& \frac{1-p}{p}\sum_{k=1}^{\infty}\frac{\Gamma(k)\Gamma(1+1/p)}{\Gamma(k+4r-2)} \cdot \frac{\Gamma(k+4r-2)}{\Gamma(k+1/p)(1/p-(4r-2))}\\[5pt] &=& \frac{(1-p)\Gamma(1+1/p)}{1-2a}\sum_{k=1}^{\infty}\frac{\Gamma(k)}{\Gamma(k+1/p)}\\[5pt] &=&\frac{(1-p)\Gamma(1+1/p)}{1-2a}\cdot \frac{\Gamma(1)}{\Gamma(1/p)(1/p-1)}=\frac{1}{1-2a}. \end{eqnarray*}
Proof of Theorem
1.4. Recall that
$r=1$
. Applying Lemma 4.4 gives
\begin{eqnarray} \frac{1}{n} \sum_{k=1}^n |W_{nk}|^{\alpha} \stackrel{\mathrm{P}}{\longrightarrow} c(\alpha,p,1),\;\qquad \frac{1}{n} \sum_{k=1}^n |W_{nk}|\log |W_{nk}| \stackrel{\mathrm{P}}{\longrightarrow} c, \end{eqnarray}
and condition (A6) holds by (4.16).
Note that
$W_{nk}\ge 0$
and
$\sum_{k=1}^n W_{nk}=n$
. It follows from (2.11) that
\begin{eqnarray} \frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k-b_n)-Y_{n,\alpha}\stackrel{\mathrm{d}}{\longrightarrow} c(\alpha,p,1)^{1/\alpha} (S-\gamma), \end{eqnarray}
where
\begin{eqnarray*} Y_{n,\alpha}= \gamma+\left\{ \begin{array}{l@{\quad}l} 0 & \mbox{if}\;\alpha\ne 1,\\[5pt] (2b\kappa/\pi)H_n & \mbox{if}\;\alpha= 1, \end{array} \right. \end{eqnarray*}
where
\begin{align*} H_n=-\frac{1}{n}\sum_{k=1}^n W_{nk} \log W_{nk}. \end{align*}
By (4.21),
$Y_{n,\alpha}\stackrel{\mathrm{P}}{\longrightarrow} c_{\alpha}$
, where
\begin{eqnarray*} c_{\alpha}\;:\!=\;\gamma+\left\{ \begin{array}{l@{\quad}l} 0 & \mbox{if}\;\alpha\ne 1,\\[5pt] -2bc\kappa/\pi & \mbox{if}\;\alpha= 1. \end{array} \right. \end{eqnarray*}
Combining this with (4.22) yields
\begin{eqnarray*} \frac{T_n-nb_n}{a_n}=\frac{1}{a_n}\sum_{k=1}^n W_{nk} (\xi_k-b_n) \stackrel{\mathrm{d}}{\longrightarrow} c(\alpha,p,1)^{1/\alpha} (S-\gamma)+c_{\alpha}. \end{eqnarray*}
Finally, from (1.13),
\begin{eqnarray*} c(\alpha,p,1)=\frac{1-p}{p}\sum_{k=1}^{\infty} k^{\alpha}\mathrm{B}(k, 1+1/p). \end{eqnarray*}
In particular, for
$\alpha=1$
, identity (8) in [Reference Bertoin10] gives
\begin{eqnarray*} c(1,p,1)=\frac{1-p}{p}\sum_{k=1}^{\infty} k\mathrm{B}(k, 1+1/p)=1. \end{eqnarray*}
This completes the proof of Theorem 1.4.
Acknowledgements
We thank two anonymous referees for their constructive comments that helped improve the quality of the paper.
Funding information
Zhishui Hu is partially supported by Hebei Natural Science Foundation (grant no. A20255 01005). Liang Dong is supported by the National Natural Science Foundation of China (grant no. 12601280).
Competing interests
There are no competing interests to declare that arose during the preparation of or the publication process for this article.
