2.1. Multiple Wiener–Itô integrals

The information recalled below about Gaussian analysis and multiple Wiener–Itô integrals can be found in [Reference Janson12]. Let $(E,\mathcal{E},\mu)$ be a $\sigma$ -finite measure space, and let W be a Gaussian (independently scattered) random measure on E with control measure $\mu$ so that $\mathbb{E} W(A)W(B)=\mu(A\cap B)$ for $A,B\in \mathcal{E}$ with $\mu(A),\mu(B)<\infty$ . Throughout this paper, the underlying probability space which carries the randomness of Gaussian random measures is denoted by $(\Omega,\mathcal{F},\mathbb{P})$ . Then, for $g\in L^2(E,\mathcal{E},\mu)$ , the Wiener integral $I(g)=\int_E g(x) W({\rm d} x)$ can be defined, which forms a linear isometry between $L^2(E,\mathcal{E},\mu)$ and the Gaussian Hilbert space $H=\{I(g)\,:\, g\in L^2(E,\mathcal{E},\mu)\}$ .

Let $H^{:p:}=\overline{\mathcal{P}}_p(H)\cap\left( \overline{\mathcal{P}}_{p-1}(H)^{\perp}\right)$ be the pth Wiener chaos of H, $p\ge 1$ , where $\overline{\mathcal{P}}_p(H)$ denotes the closure of the linear subspace of multivariate polynomials of elements of H with degrees p or lower, and $\perp$ in the superscript denotes the orthogonal complement in $L^2(\Omega,\mathcal{F},\mathbb{P})$ . Then, for $f\in L^2(E^p,\mathcal{E}^p,\mu^p)$ , the p-tuple Wiener–Itô integral

\begin{equation*}I_p(f)=\int^{\prime}_{E^p} f(x_1,\ldots,x_p) W({\rm d} x_1)\cdots W({\rm d} x_p)\end{equation*}

can be defined [Reference Janson12, Theorem 7.26]. In fact, $I_p\,:\,L^2(E^p,\mathcal{E}^p,\mu^p) \rightarrow H^{:p:}$ forms a bounded linear operator, and in particular a linear isometry from the subspace of symmetric functions of $L^2(E^p,\mathcal{E}^p,\mu^p/p!)$ to $H^{:p:}$ . In addition, $I_p$ is characterized by the following property: for $f_1,\ldots,f_p\in L^2(E,\mathcal{E},\mu)$ , we have

(2.1) \begin{equation}I_p(f_1\otimes\cdots\otimes f_p)=\int^{\prime}_{E^p} f_1(x_1)\cdots f_p(x_p) W({\rm d} x_1)\cdots W({\rm d} x_p)= {:}\, I(f_1)\cdots I(f_p)\,{:},\end{equation}

where for $\xi_1,\ldots,\xi_p\in H$ , the notation ${:}\,\xi_1 \cdots \xi_p\,{:}$ stands for the Wick product, which is the projection of the product $\xi_1\cdots \xi_p$ onto the $L^2$ subspace $H^{:p:}$ . We note that in the literature the construction of $I_p(f)$ often starts with (2.1) for $f_i=1_{A_i}$ for disjoint $A_i\in \mathcal{F}$ with $\mu(A_i)<\infty$ , so that ${:}\, I(f_1)\cdots I(f_p)\,{:}$ is simply the product $W(A_1)\cdots W(A_p)$ . Then, the definition is extended to a general $f\in L^2(E^p,\mathcal{E}^p,\mu^p)$ by linearity and continuity given that $\mu$ is atomless. If $\mu$ has atoms, an extra step in the construction is needed (see, e.g., [Reference Major18, p. 42]). For generality, we shall use the construction of multiple Wiener–Itô integrals in [Reference Janson12, Chapter VII.2] without assuming that $\mu$ is atomless.

The following lemma is useful for changing the underlying measure spaces in multiple Wiener–Itô integral representations of a process.

Lemma 2.1. Let $(E_j,\mathcal{E}_j,\mu_j)$ , $j=1,2$ , be $\sigma$ -finite measure spaces. Suppose $W_j$ is a Gaussian random measure defined on $E_j$ with control measure $\mu_j$ , $j=1,2$ . Let $(U,\mathcal{H})$ be a measurable space and suppose $f_j:E_j \rightarrow U $ , $j=1,2$ , are measurable and satisfy

(2.2) \begin{equation}\mu_1 \circ f_1^{-1} = \mu_2 \circ f_2^{-1}=\!:\,\nu \end{equation}

on $(U,\mathcal{H})$ . Let T be an index set and let $g_t\,:\, U^p\rightarrow[-\infty,+\infty]$ , $t\in T$ , be a family of measurable functions satisfying $g_t\circ f_j^{\otimes p} \in L^2(E^p_j,\mathcal{E}_j^p, \mu_j^p)$ , $j=1,2$ . Let

(2.3) \begin{equation}Z_j(t)=\int_{E_j^p}^{\prime} g_t\left(f_j(x_1),\ldots,f_j(x_p)\right) W_j({\rm d} x_1)\cdots W_j({\rm d} x_p), \qquad j=1,2.\end{equation}

Then, the two processes $(Z_j(t))_{t\in T}$ , $j=1,2$ , have the same finite-dimensional distributions.

Proof. First, by Cramér–Wold and the linearity of the multiple integrals, it suffices to prove equality of marginal distributions at a single $t\in T$ , and we thus simply set $g_t=g$ . Suppose first $g=h_1\otimes \cdots \otimes h_p$ , where $h_1,\ldots,h_p\in L^2(U,\mathcal{H},\nu)$ . Then, by (2.1), the right-hand side of (2.3) is equal to

\begin{equation*}{:}\int_{E_j} h_1 \circ f_j (x) W_j({\rm d} x) \cdots \int_{E_j} h_p \circ f_j (x) W_j({\rm d} x)\,{:}, \qquad j=1,2.\end{equation*}

So, by joint Gaussianity, the conclusion follows from comparing the covariances: for $1\le i_1, i_2\le p$ ,

\begin{align*}&\mathbb{E} \bigg[\int_{E_j} h_{i_1} \circ f_j (u) W_j({\rm d} x) \int_{E_j} h_{i_2} \circ f_j (x) W_j({\rm d} x)\bigg] \\[3pt] &\quad= \int h_{i_1}\circ f_j(x) \ h_{i_2}\circ f_j(x)\mu_j({\rm d} x)= \int_{U} h_{i_1}(u) h_{i_2}(u)\nu({\rm d} u),\qquad j=1,2,\end{align*}

where we have used (2.2) in the last equality.

Similarly, using the linearity of the multiple integrals, the conclusion holds if g is a finite linear combination of terms each of the form $h_1\otimes\cdots\otimes h_p$ . Finally, by the linear isometry of the multiple integral, it suffices to note that such linear combinations are dense in $L^2(U^p, \mathcal{H}^p,\nu^p)$ . □

We mention that a similar discussion to the above carries over to the case where W is replaced by a complex-valued Gaussian random measure (see [Reference Janson12, Chapter 7, Section 4]).

2.2. Regenerative sets and local time functional

Most of the information reviewed in this section about subordinators and regenerative sets can be found in [Reference Bertoin4].

Recall that a process $(\sigma(t))_{t\ge0}$ is said to be a subordinator if it is a nondecreasing Lévy process starting at the origin. The Laplace exponent of $(\sigma(t))_{t\ge0}$ is given by $\mathbb{E} {\rm e}^{-\lambda \sigma(t)}=\exp({-} t\Phi(\lambda))$ , $\lambda\ge0$ , which completely characterizes its law and satisfies the Lévy–Khintchine formula,

(2.4) \begin{equation}\Phi(\lambda)= {\rm d}\lambda+\int_{(0,\infty)}(1-{\rm e}^{-\lambda x})\Pi({\rm d} x),\end{equation}

where the constant $d\ge 0$ is the drift and $\Pi$ is the Lévy measure on $(0,\infty)$ satisfying $\int_{(0,\infty)} (x\wedge 1) \nu({\rm d} x)<\infty$ . The renewal measure U of $(\sigma(t))_{t\ge 0}$ on $[0,\infty)$ is characterized by $\int_{[0,\infty)}f(x)U({\rm d} x)=\mathbb{E} \int_0^\infty f(\sigma(t))\,{\rm d} t$ for any measurable function $f\ge 0$ , which is related to the Laplace exponent through

(2.5) \begin{equation}\int_{[0,\infty)} {\rm e}^{-\lambda x} U({\rm d} x)=\frac{1}{\Phi(\lambda)}, \qquad \lambda\ge 0.\end{equation}

The Radon–Nikodym derivative of U with respect to the Lebesgue measure, if it exists, is called the renewal density.

Let $\textbf{{F}}=\textbf{{F}}([0,\infty))$ denote the space of closed subsets of $[0,\infty)$ equipped with the Fell topology [Reference Molchanov19, Appendix C]. A random element R taking values in $\textbf{F}$ is said to be a regenerative set if R has the same distribution as the closed range $\overline{\{\sigma(t):\ t\ge 0\}}$ , where $(\sigma(t))_{t\ge0}$ is a subordinator. Note that for any constant $c>0$ , the time-scaled subordinator $(\sigma(ct))_{t\ge 0}$ and the original $(\sigma(t))_{t\ge 0}$ correspond to the same regenerative set. To make the correspondence between a regenerative set and a subordinator unique, we shall always assume the following normalization condition for the Laplace exponent of the subordinator:

(2.6) \begin{equation}\Phi(1)=1.\end{equation}

A regenerative set R is said to be $\beta$ -stable, $\beta\in (0,1)$ , if the associated subordinator $(\sigma (t))_{t\ge 0}$ is $\beta$ -stable, namely, if the Laplace exponent $\Phi(\lambda)=\lambda^{\beta}$ , which corresponds to the Lévy measure

(2.7) \begin{equation}\Pi ({\rm d} x)=\frac{ \beta}{\Gamma(1-\beta)}x^{-1-\beta} {1}_{(0,\infty)}(x) \,{\rm d} x.\end{equation}

A random closed set F in $\textbf{F}$ is said to be stationary if

(2.8) \begin{equation}\tau_x (F)\,:\!=F\cap [x,\infty)-x\overset{{\rm d}}{=} F\end{equation}

for any $x\ge 0$ . A $\beta$ -stable regenerative set R itself is not stationary. However, stationarity in a generalized sense can be obtained by a shifted R as follows.

Proposition 2.1. Let $\mathbb{P}_R$ be the distribution of a $\beta$ -stable regenerative set on $\textbf{F}$ and let $\pi_V$ be a Borel measure on $[0,\infty)$ with $\pi_V({\rm d} v)=(1-\beta) v^{-\beta} {\rm d} v$ , $v\in (0,\infty)$ . There exists a $\sigma$ -finite infinite-measure space $(E,\mathcal{E},\mu)$ and a measurable mapping $ \overline{R}\,:\,E\rightarrow\textbf{{F}}$ such that

\begin{equation*}\mu ( \overline{R} \in \cdot )=(\mathbb{P}_R\times \pi_V)(F+v\in \cdot ),\end{equation*}

where the right-hand side is understood as the push-forward measure of $\mathbb{P}_R\times \pi_V$ under the mapping $(F,v)\mapsto F+v$ . Furthermore, $ \overline{R} $ is stationary in the sense of

\begin{equation*}\mu(\tau_x \overline{R} \in \cdot )= \mu( \overline{R} \in \cdot ),\end{equation*}

where $\tau_x$ is as in (2.8).

Next, we recall the local time functionals introduced by [Reference Kingman14]. For $\beta\in (0,1)$ , define

\begin{equation*}L^{(\beta)}\,:\, {F}\rightarrow [0,\infty], \quad L^{(\beta)}(F) \,:\!=\limsup_{n\rightarrow\infty}\frac{\lambda\left(F+[- \frac{1}{2n},\frac{1}{2n}]\right)}{ l_{\beta}(n)},\end{equation*}

where $\lambda$ is the Lebesgue measure, $F+[- 1/{2n},1/{2n}]=\cup_{x\in F}[x-1/{2n},x+1/{2n}]$ , and the normalization sequence

\begin{equation*}l_{\beta}(n)=\int_{0}^{1/n} \Pi((x,\infty))\,{\rm d} x = \frac{ n^{\beta-1}}{\Gamma(2-\beta)},\end{equation*}

where $\Pi$ is as in (2.7). We then define

(2.9) \begin{equation}L_t^{(\beta)}(F)\,:\!=L^{(\beta)}(F\cap [0,t]), \qquad t\ge 0.\end{equation}

By [Reference Bai, Owada and Wang1, Lemma 2.1], $L_t^{(\beta)}$ is a measurable mapping from $\textbf{F}$ to $[0,\infty]$ for each $t\ge 0$ . Furthermore, [Reference Kingman14, Theorem 3] entails that if R is a $\beta$ -stable regenerative set, then the process $(L_t^{(\beta)}(R ))_{t\ge 0}$ has the same finite-dimensional distributions as the local time process associated with R, which is an inverse $\beta$ -stable subordinator.

4.1. Constructions of regenerative sets and local times by random covering

The proof of Theorem 3 uses a construction of a regenerative set as the set left uncovered by random intervals due to [Reference Fitzsimmons, Fristedt and Shepp9], as well as a related construction of local time which originates from [Reference Bertoin and Pitman5]. Similar constructions are used in [Reference Bai, Owada and Wang1].

Suppose a Poisson point process $\mathcal{N}=\sum_{\ell\ge 1} \delta_{\{y_\ell,z_\ell\}}$ on $[0,\infty)^2$ is defined on the probability space $(H,\mathcal{H},\mathbb{P}_\mathcal{N})$ with intensity measure $(1-\beta) \,{\rm d} y z^{-2}\,{\rm d} z$ , $\beta\in (0,1)$ , where $(\{y_\ell,z_\ell\})_{\ell\ge 1}$ is a measurable enumeration of the points of $\mathcal{N}$ . For $\varepsilon\ge 0$ , define

(4.2) \begin{equation}R_\varepsilon = \bigcap_{\ell:\ z_\ell\ge \varepsilon}( y_\ell,y_\ell+z_\ell)^{\rm c},\end{equation}

where the complement is with respect to $[0,\infty)$ . In other words, $R_\varepsilon $ is the subset of $[0,\infty)$ left uncovered by the collection of (possibly overlapping) open intervals $\{ (y_\ell,y_\ell+z_\ell)\,:\, z_\ell\ge \varepsilon\}$ , where the left boundary $y_\ell$ and the width $z_\ell$ are governed by $\mathcal{N}$ . In view of [Reference Fitzsimmons, Fristedt and Shepp9, Example 1], each $R_\varepsilon $ , $\varepsilon\ge 0$ , is a regenerative set on $[0,\infty)$ , and in particular, $R_0$ is a $\beta$ -stable regenerative set. It is also known that the subordinator associated with $R_\varepsilon$ has a positive drift $d_\varepsilon$ when $\varepsilon>0$ . In contrast to $R_0$ , the Lévy measure $\Pi_\varepsilon$ and the drift $d_\varepsilon$ of $R_\varepsilon$ , $\varepsilon>0$ , do not have simple expressions, although the exact expressions are not needed for our purpose. While other constructions of regenerative sets are possible, the random covering scheme seems efficient in providing tractable approximation of the intersecting regenerative sets and the local times encountered in the proof of Theorem 3.1.

By the calculation in the proof of [Reference Bai, Owada and Wang1, Lemma 2.5], we have, for $\varepsilon>0$ ,

(4.3) \begin{equation}p_\varepsilon(x)\,:\!=\mathbb{P}_\mathcal{N}(x\in R_{\varepsilon}) = {\rm e}^{(\beta-1)x/\varepsilon}{1}_{\{0\le x\le \varepsilon\}}+ \left(\frac{\varepsilon}{{\rm e}}\right)^{1-\beta} x^{\beta-1}{1}_{\{x>\varepsilon\}},\qquad x\ge 0.\end{equation}

Let $u_\varepsilon(x)$ be the renewal density of the subordinator associated with $R_\varepsilon$ . By [Reference Bertoin4, Propositions 1.7], if $\varepsilon>0$ ,

(4.4) \begin{equation}u_\varepsilon(x)= d_{\varepsilon}^{-1}p_\varepsilon(x)=\!:\,d_{\varepsilon}^{-1}\left(\frac{\varepsilon}{{\rm e}}\right)^{1-\beta}f_\varepsilon(x), \qquad x>0.\end{equation}

It is elementary to verify that as $\varepsilon\downarrow 0$ ,

(4.5) \begin{equation}f_\varepsilon(x)= ({\rm e}^{x/\varepsilon-1}\varepsilon)^{\beta-1}{1}_{\{0\le x\le \varepsilon\}}+ x^{\beta-1}{1}_{\{x>\varepsilon\}}\uparrow x^{\beta-1},\qquad x>0.\end{equation}

Note that $\int_{0}^\infty {\rm e}^{-x} u_\varepsilon(x)\,{\rm d} x = 1$ due to normalization requirement (2.6) via (2.5). Combining this fact with (4.5) and the monotone convergence theorem, we have, as $\varepsilon\rightarrow 0$ ,

(4.6) \begin{equation}d_\varepsilon\sim \Gamma(\beta)\left(\frac{\varepsilon}{{\rm e}}\right)^{1-\beta} \quad \text{as }\varepsilon\rightarrow 0.\end{equation}

Note that $d_0=0$ since a $\beta$ -stable subordinator has no drift. Define, for all $\varepsilon\ge 0$ , the measures

(4.7) \begin{equation}\pi_\varepsilon({\rm d} x)=d_\varepsilon \delta_0 + \overline{\Pi}_\varepsilon (x)\,{\rm d} x,\quad x\ge 0,\qquad \widetilde{\pi}_\varepsilon(\cdot )=\frac{\pi_\varepsilon(\cdot\cap [0,1])}{\pi_\varepsilon([0,1])},\end{equation}

where $\overline{\Pi}_{\varepsilon}(x)=\Pi_\varepsilon((x,\infty))$ is the tail of the Lévy measure $\Pi_\varepsilon$ corresponding to $R_\varepsilon $ , and $\delta_0$ is the delta measure with a unit mass at 0. Note that $\Pi_0$ is equal to $\Pi$ in (2.7) and $\widetilde{\pi}_0=\pi_V$ in Proposition 2.1 when restricted to [0, 1]. Note that, for each $\varepsilon\ge 0$ ,

\begin{equation*}\int_0^y u_\varepsilon(x)\,{\rm d} x\asymp y^{\beta}, \qquad y>0,\end{equation*}

where $h_1(y)\asymp h_2(y)$ means $ c_1 h_2(y)\le h_1(y)\le c_2 h_2(y)$ for some constants $0<c_1<c_2$ . So, in view of [Reference Bertoin4, Proposition 1.4], we have

\begin{equation*}\int_0^y \overline{\Pi}_{\varepsilon}(x)\,{\rm d} x \asymp y^{1-\beta}, \qquad y>0.\end{equation*}

Hence, each $\pi_\varepsilon$ is a $\sigma$ -finite infinite measure on $[0,\infty)$ .

Enriching the space $(H,\mathcal{H})$ if necessary, suppose $\nu$ is a $\sigma$ -finite measure on $\mathcal{H}$ , and suppose $U_\varepsilon\,:\,H\rightarrow[0,\infty)$ is an improper random variable satisfying

(4.8) \begin{equation}\nu((R_\varepsilon,U_\varepsilon)\in\cdot)=(\mathbb{P}_{R_\varepsilon}\times \pi_\varepsilon)(\cdot),\end{equation}

where $\mathbb{P}_{R_\varepsilon}$ is the distribution of $R_\varepsilon$ on $\textbf{F}$ and $\pi_\varepsilon$ is as in (4.7). Then, in view of [Reference Fitzsimmons and Taksar10], the improper random set $\overline{R}_\varepsilon\,:\!=R_\varepsilon+U_\varepsilon$ is stationary in the sense of

(4.9) \begin{equation}\nu(\tau_x \overline{R}_\varepsilon \in \cdot )= \nu( \overline{R}_\varepsilon \in \cdot ),\end{equation}

where $\tau_x$ is the shift as in (2.8).

The next result enables a key coupling argument in the proof of Theorem 3.1.

Lemma 4.1. We have the convergence in total variation distance:

\begin{equation*}\|\widetilde{\pi}_\varepsilon- \widetilde{\pi}_0\|_{\rm TV}\,:\!=\sup\{|\widetilde{\pi}_\varepsilon(A)-\widetilde{\pi}_0(A)|\,:\, A\in \mathcal{B}([0,1])\}\rightarrow 0 \qquad {as } \varepsilon\rightarrow 0.\end{equation*}

Proof. We first show that, as $\varepsilon\rightarrow 0$ ,

(4.10) \begin{equation} \pi_\varepsilon([0,1])\rightarrow \pi_0([0,1]).\end{equation}

By [Reference Fitzsimmons, Fristedt and Maisonneuve8, Equation (1.11) and Proposition 3.9], as $\varepsilon\rightarrow 0$ , the Laplace exponent uniquely associated with $R_{\varepsilon}$ satisfying (2.6) converges pointwise to the Laplace exponent uniquely associated with a $\beta$ -stable regenerative set. This, by [Reference Kallenberg13, Theorem 15.15(ii)], further implies that as $\varepsilon\rightarrow 0$ , we have the following weak convergence of measures:

(4.11) \begin{align}\widetilde{\nu}_\varepsilon({\rm d} x)&\,:\!= d_\varepsilon \delta_0 +(1-{\rm e}^{-x})\Pi_{\varepsilon}({\rm d} x) \notag\\[3pt] &\overset{{\rm d}}{\rightarrow} (1-{\rm e}^{-x})\Pi_0({\rm d} x)=(1-{\rm e}^{-x}) \frac{\beta}{\Gamma(1-\beta)} x^{-\beta-1} \, {\rm d} x =\!:\,\widetilde{\nu}_0({\rm d} x),\end{align}

where $\widetilde{\nu}_\varepsilon$ , $\varepsilon\ge0$ , are probability measures on $[0,\infty)$ due to (2.4) and (2.6). Next, by Fubini,

(4.12) \begin{align}\pi_\varepsilon([0,1])=d_\varepsilon+\int_0^1 \overline{\Pi}_\varepsilon(x)\,{\rm d} x = d_\varepsilon+\int_{(0,\infty)}(1\wedge x)\Pi_\varepsilon({\rm d} x) = \int_{[0,\infty)} h(x) \widetilde{\nu}_\varepsilon({\rm d} x),\end{align}

where $h(x)\,:\!=\frac{1\wedge x}{1-{\rm e}^{-x}}$ for $x>0$ and $h(0)\,:\!=1$ is a bounded continuous function on $[0,\infty)$ . Hence, by the weak convergence in (4.11), as $\varepsilon \rightarrow 0$ we have

(4.13) \begin{equation}\int_{[0,\infty)} h(x) \widetilde{\nu}_\varepsilon({\rm d} x)\rightarrow \int_{[0,\infty)} h(x) \widetilde{\nu}_0({\rm d} x) =\int_0^1 \overline{\Pi}_0(x)\,{\rm d} x=\pi_0([0,1])=\frac{1}{\Gamma(2-\beta)}.\end{equation}

Combining (4.12) and (4.13), we obtain (4.10).

Now, to conclude the proof, it suffices to show that

(4.14) \begin{equation}\sup\{| \pi_\varepsilon(A)-\pi_0(A)|\,:\, A\in \mathcal{B}([0,1])\}\rightarrow 0 \qquad \text{as } \varepsilon\rightarrow 0.\end{equation}

Indeed, for $A\in \mathcal{B}([0,1])$ ,

(4.15) \begin{align} |\pi_\varepsilon(A)-\pi_0(A)| & = \bigg| d_\varepsilon \delta_0(A) + \int_{A\cap (0,1]} \overline{\Pi}_\varepsilon(x) - \overline{\Pi}_0(x) \,{\rm d} x\bigg| \nonumber \\ & \le d_\varepsilon+ \int_{[0,1]} \big|\overline{\Pi}_\varepsilon(x) - \overline{\Pi}_0(x)\big|\,{\rm d} x.\end{align}

Note that $d_\varepsilon\rightarrow 0$ by (4.6). On the other hand, [Reference Fitzsimmons, Fristedt and Maisonneuve8, Proposition 3.9] and [Reference Kallenberg13, Lemma 15.14(ii)] imply that, as $\varepsilon\rightarrow 0$ , $\overline{\Pi}_\varepsilon(x) \rightarrow \overline{\Pi}_0(x)$ , $x>0$ . In addition, by (4.12), (4.13), and the fact $d_\varepsilon\rightarrow 0$ , we have, as $\varepsilon\rightarrow 0$ ,

\begin{equation*}\int_0^1 \overline{\Pi}_\varepsilon(x)\,{\rm d} x \rightarrow \int_0^1 \overline{\Pi}_0(x)\,{\rm d} x.\end{equation*}

Therefore, the second term in the bound in (4.15) tends to 0 as $\varepsilon\rightarrow 0$ by Scheffé’s lemma (e.g. [Reference Williams33, Item 5.10]). So, (4.14) follows. □

Next, we turn to the construction of the local time based on the covering scheme introduced above. Suppose $R_\varepsilon$ , $\varepsilon\ge 0$ , are as in (4.2) with the point process $\mathcal{N}$ defined on a probability space $(H,\mathcal{H},\mathbb{P}_\mathcal{N})$ . In view of Lemma 4.1 and a well-known coupling characterization of total variation distance (e.g. [Reference Sethuraman26, Theorem 2.1]), there exist random variables $V_\varepsilon\ge 0$ , $\varepsilon\ge 0$ , defined on a probability space $(\Theta,\mathcal{G},\mathbb{P}_V)$ , such that $\mathbb{P}_V(V_\varepsilon\in\cdot)=\widetilde{\pi}_\varepsilon(\cdot)$ and, as $\varepsilon\rightarrow 0$ ,

(4.16) \begin{equation}\mathbb{P}_V(V_\varepsilon = V_0)\rightarrow 1.\end{equation}

Lemma 4.2. For $0\le t\le 1$ , set

(4.17) \begin{equation}L_t^{(\varepsilon)}(\textbf{{h}},\boldsymbol{\theta})=\frac{1}{\Gamma(\beta_p)} \left(\frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \int_0^t {1}_{\{x\in \bigcap_{i=1}^p (R_\varepsilon(h_i)+V_{\varepsilon}(\theta_i)) \}} \,{\rm d} x,\end{equation}

where $\textbf{{h}}\,:\!=(h_1,\ldots,h_p)\in H^p$ , $\boldsymbol{\theta}\,:\!=(\theta_1,\ldots,\theta_p)\in \Theta^p$ , and $\beta_p$ is as in (3.1).

1. (a) As $\varepsilon =1/n \rightarrow 0$ , $n\in \mathbb{Z}_+$ , $L_{t}^{(\varepsilon)}$ converges in $L^r=L^r\big(H^p\times \Theta^p,\mathcal{H}^{p}\times \mathcal{G}^p , \mathbb{P}_\mathcal{N}^p\times \mathbb{P}_V^p\big)$ to a limit $L_t^*$ , $r\in \mathbb{Z}_+$ .

2. (b) Let $\mathbb{E}^{\prime}$ denote integration with respect to $ \mathbb{P}_\mathcal{N}^p\times \mathbb{P}_V^p$ . Then, for $r\in \mathbb{Z}_+$ ,

   (4.18) \begin{equation}\mathbb{E}^{\prime} [(L_{t}^*)^r] = \frac{r! \Gamma(2-\beta)^p\Gamma(\beta)^p}{\Gamma((r-1)\beta_p+2)\Gamma(\beta_p)} \, t^{(r-1)\beta_p-1}.\end{equation}

3. (c) We have

   (4.19) \begin{equation}L_t^*(\textbf{{h}},\boldsymbol{\theta})= L_t\Bigg(\bigcap_{i=1}^p R_0(h_i ) +V_0(\theta_{i})\Bigg) \quad \mathbb{P}_\mathcal{N}^p\times \mathbb{P}_V^p\text{-a.e. (almost everywhere)},\end{equation}
   where $L_t=L_t^{(\beta_p)}$ is the local time functional as in (2.9).

We need the following preparation for the proof of the lemma. Note that throughout, an integral $\int_a^b \cdot \ {\rm d} x$ is understood as zero if $a\ge b$ .

Lemma 4.3. For $\delta,\eta\ge 0 $ , $\textbf{{v}}=(v_1,\ldots,v_p)\in [0,1]^p$ and $\textbf{{h}}=(h_1,\ldots,h_p)\in H^p$ , define

\begin{equation*}\Delta_{\delta,\eta}^{(\varepsilon)}(\textbf{{h}};\,\textbf{{v}})=\frac{1}{\Gamma(\beta_p)} \left(\frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \int_\delta^\eta {1}_{\{x\in \bigcap_{i=1}^p (R_\varepsilon(h_i)+v_i) \}} \, {\rm d} x .\end{equation*}

Let $\mathbb{E}_\mathcal{N}$ denote integration with respect to $\mathbb{P}_\mathcal{N}^p$ on $H^p$ , and suppose $r\in \mathbb{Z}_+$ . Then

(4.20) \begin{align} &\mathbb{E}_{\mathcal{N}} \big[ \Delta_{\delta,\eta}^{(\varepsilon)}( \cdot \, ;\,\textbf{{v}})^r\big] \nonumber \\ & \quad= \frac{r!}{\Gamma(\beta_p)^r}\int_{\delta<x_1<\cdots<x_r<\eta} \Bigg(\prod_{i=1}^p f_\varepsilon(x_1-v_i) \Bigg)\, f_\varepsilon(x_2-x_1)^{p} \cdots f_{\varepsilon}(x_r-x_{r-1})^{p} \, {\rm d}\textbf{{x}} \end{align}

(4.21) \begin{equation} \hspace*{-258pt}\le c (\eta-\delta)_+^{r\beta_p}, \end{equation}

where $f_\varepsilon(x)$ is as in (4.5) if $x\ge 0$ , $f_\varepsilon(x)\,:\!=0$ for $x<0$ , and $c>0$ is a constant which does not depend on $\varepsilon$ , $\delta$ , $\eta$ , or $\textbf{{v}}$ .

Proof. Suppose $\delta< \eta$ , otherwise the conclusion is trivial. Let $p_\varepsilon(x)$ be as in (4.3) when $x\ge 0$ , and set $p_\varepsilon(x)=0$ if $x<0$ . For $0\le x_1<\cdots<x_r$ and $\varepsilon>0$ , we have

\begin{align*}&\mathbb{P}_\mathcal{N}^p\bigg(\bigg\{\textbf{{h}}\in H^p \,:\, \{x_1,\ldots,x_r\}\subset \bigcap_{i=1}^p (R_\varepsilon(h_i)+v_i)\bigg\}\bigg)\\&\quad= \prod_{i=1}^p \mathbb{P}_\mathcal{N}(\{h\in H\,:\, \{x_1,\ldots,x_r\}\subset (R_\varepsilon(h)+v_i)\})\\&\quad=\prod_{i=1}^p \big( p_\varepsilon(x_1-v_i)p_\varepsilon(x_2-x_1) \ldots p_{\varepsilon}(x_r-x_{r-1}) \big),\end{align*}

where for the last equality we have used the regenerative property of $R_\varepsilon$ [Reference Fitzsimmons, Fristedt and Shepp9, Equation (6)]. See also the proof of [Reference Bai, Owada and Wang1, Lemma 2.5]. Then, (4.20) follows from Fubini, the relation $f_\varepsilon(x)= (\varepsilon/{\rm e})^{\beta-1}p_\varepsilon(x)$ , see (4.4), and a symmetry in the multiple integral.

Next, from (4.5) we have. for any $\varepsilon>0$ ,

(4.22) \begin{equation}f_\varepsilon(x)\le x^{\beta-1}, \qquad x>0 . \end{equation}

Hence, for some constants $c_1,c_2>0$ free of $\varepsilon$ , $\delta$ , $\eta$ , or $\textbf{{v}}$ (recall that $\beta_p=p(\beta-1)+1\in (0,1)$ ),

\begin{align*}\mathbb{E}_{\mathcal{N}} & \Delta_{\delta,\eta}^{(\varepsilon)}( \cdot\, ;\,\textbf{{v}})^r \\ & \le c_1\int_{\delta<x_1<\cdots<x_r<\eta} \Bigg(\prod_{i=1}^p (x_1-v_i)_+^{\beta-1} \Bigg) (x_2-x_1)^{\beta_p-1} \cdots (x_r-x_{r-1})^{\beta_p-1} \, {\rm d}\textbf{{x}}\\& = c_2 \int_{\delta}^{\eta} \Bigg(\prod_{i=1}^p (x_1-v_i)_+^{\beta-1} \Bigg) (\eta-x_1)^{(r-1)\beta_p} \, {\rm d} x_1 =\!:\, c_2 g_{\delta,\eta}(v_1,\ldots,v_p),\end{align*}

where for the first equality we have integrated out the variables in the order $x_r$ , $x_{r-1}$ , …, $x_2$ and repeatedly applied (4.1). Note that the function $g_{\delta,\eta}\,:\,[0,1]^p\rightarrow \mathbb{R}$ is symmetric, so we suppose without loss of generality that $0\le v_1\le \cdots \le v_p \le 1$ . Then, by monotonicity and (4.1),

\begin{align*}g_{\delta,\eta}(v_1,\ldots,v_p) & = \int_{\delta}^{\eta} \Bigg(\prod_{i=1}^p (x-v_i)^{\beta-1}_+ \Bigg) (\eta-x)^{(r-1)\beta_p} \, {\rm d} x \\ & \le \int_{\delta}^{\eta} (x-v_p)_+^{\beta_p-1} (\eta-x)^{(r-1)\beta_p} \, {\rm d} x \notag \\ & \le {1}_{\{v_p\ge \delta\}} \mathrm{B}(\beta_p,(r-1)\beta_p+1) (\eta-v_p)_+^{r\beta_p} \\ & \quad + {1}_{\{v_p < \delta\}}\int_{\delta}^{\eta} (x-\delta )^{\beta_p-1} (\eta-x)^{(r-1)\beta_p} \, {\rm d} x \notag \\ & \le c (\eta-\delta)^{r\beta_p}.\\[-35pt] \end{align*}

□

Proof of Lemma 4.2. All conclusions trivially hold if $t=0$ . Suppose $0< t\le 1$ .

1. (a) Let $\Theta_*\,:\!= \bigcup_{n=1}^\infty \{\theta\,:\, V_{1/n}(\theta)=V_0(\theta)\}$ ; we have $\mathbb{P}_V(\Theta_*)=1$ by (4.16). Let $V_0^{\rm M}(\boldsymbol\theta)= \max(V_0(\theta_i),\ i=1,\ldots,p)$ . Set $D_1=\{\boldsymbol{\theta}\in \Theta^p_*\,:\, 0<V_0^{\rm M}(\boldsymbol\theta)<t \}$ , $D_2=\{\boldsymbol{\theta}\in \Theta^p_*\,: V_0^{\rm M}(\boldsymbol\theta)>t\}$ , $E_1= H^p \times D_1$ , and $E_2=H^p \times D_2$ , which satisfy $\big(\mathbb{P}_\mathcal{N}^p\times \mathbb{P}_V^p\big)(E_1\cup E_2) =1$ because the distribution $\widetilde{\pi}_0$ of $V_0$ is continuous. We shall establish the $L^r$ convergence restricted on $E_1$ and $E_2$ respectively. Let $\mathbb{E}_{\mathcal{N}}$ and $\mathbb{E}_{V}$ denote the integrations with respect to $\mathbb{P}_{\mathcal{N}}^p$ and $\mathbb{P}_V^p$ respectively.

   First, suppose $\boldsymbol{\theta}\in D_1$ . In this case, since $V_0^{\rm M}(\boldsymbol{\theta})\ge \inf \big(\cap_{i=1}^p R_\varepsilon(h_i)+V_{0}(\theta_i)\big)$ ,

   \begin{align*}L_t^{(\varepsilon)}(\textbf{{h}},\boldsymbol{\theta})=\frac{1}{\Gamma(\beta_p)}\left(\frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \int_{V_0^{\rm M}(\boldsymbol{\theta})}^t {1}_{\{x\in \bigcap_{i=1}^p (R_\varepsilon(h_i)+V_{\varepsilon}(\theta_i)) \}} \, {\rm d} x.\end{align*}
   For $m\in \mathbb{Z}_+$ , set $\delta_m(\boldsymbol{\theta})= (1-m^{-1})V_0^{\rm M}(\boldsymbol{\theta})+m^{-1}t$ , which satisfies $0 <V_0^{\rm M}(\boldsymbol{\theta}) <\delta_m(\boldsymbol{\theta})<t$ . Define
   (4.23) \begin{equation}L_t^{(\varepsilon,m)}(\textbf{{h}},\boldsymbol{\theta}) = \frac{1}{\Gamma(\beta_p)} \left(\frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \int_{\delta_m(\boldsymbol{\theta})}^t {1}_{\{x\in \bigcap_{i=1}^p ( R_\varepsilon(h_i)+V_{\varepsilon}(\theta_i)) \}} \, {\rm d} x \le L_{t}^{(\varepsilon)}(\textbf{{h}},\boldsymbol{\theta}).\end{equation}
   If $\varepsilon=1/n$ is sufficiently small that $V_\varepsilon(\theta_i)=V_0(\theta_i)$ for all $i=1,\ldots,p$ and $\varepsilon<\delta_m(\boldsymbol{\theta})-V_0^{\rm M}(\boldsymbol{\theta})$ , by [Reference Bai, Owada and Wang1, Lemma 2.6], $(L_t^{(\varepsilon,m)}(\cdot\,,\boldsymbol{\theta}))_{\varepsilon>0} $ forms a nonnegative $\mathbb{P}_\mathcal{N}^p$ -martingale as $\varepsilon$ decreases. So, $L_t^{(\varepsilon,m)}(\cdot\,,\boldsymbol{\theta}) $ converges $\mathbb{P}_{\mathcal{N}}^p$ -a.e. as $\varepsilon=1/n\rightarrow 0$ , and hence $L_t^{(\varepsilon,m)}$ converges $\mathbb{P}_{\mathcal{N}}^p\times \mathbb{P}_{V}^p$ -a.e. by Fubini. On the other hand, for any $r\in \mathbb{Z}_+$ , by Fubini, (4.21), and (4.23),
   (4.24) \begin{equation}\mathbb{E}^{\prime} \big|L_t^{(\varepsilon,m)}\big|^r \le \mathbb{E}_{V}\mathbb{E}_\mathcal{N}\big[(L_t^{(\varepsilon)})^r\big] \le c t^{r\beta_p}.\end{equation}
   Hence, the $L^r$ convergence of $L_t^{(\varepsilon,m)}1_{E_1}$ as $\varepsilon=1/n\rightarrow 0$ follows from uniform integrability. On the other hand, by (4.21) again,
   \begin{equation*} \mathbb{E}_{\mathcal{N}} \big|L_{t}^{(\varepsilon)}(\cdot\,,\boldsymbol{\theta})- L_{t}^{(\varepsilon,m)}(\cdot\, ,\boldsymbol{\theta})\big|^r \le c \big[\delta_m(\boldsymbol{\theta})-V_0^{\rm M}(\boldsymbol{\theta})\big]^{r\beta_p} \le c t m^{-r\beta_p}.\end{equation*}
   Therefore, $\lim_{m\rightarrow \infty}\sup_{\varepsilon>0} \big\|L_{t}^{(\varepsilon)}{1}_{E_1}- L_{t}^{(\varepsilon,m)} {1}_{E_1}\big\|_{L^r}=0$ . This, together with the $L^r$ convergence of $L_{t}^{(\varepsilon,m)}{1}_{E_1}$ as $\varepsilon=1/n\rightarrow 0$ implies that $L_{t}^{(\varepsilon)}{1}_{E_1}$ is Cauchy in $L^r$ as $\varepsilon=1/n\rightarrow 0$ , and thus converges in $L^r$ .

   Next, suppose $\boldsymbol{\theta}\in D_2$ . When $\varepsilon$ is small enough that $V_{\varepsilon}(\theta_i)=V_0(\theta_i)$ for all $i=1,\ldots,p$ , we have $L_{t}^{(\varepsilon)}(\textbf{{h}},\boldsymbol{\theta})=0$ since $t\le \inf\big(\cap_{i=1}^p R_\varepsilon(h_i)+V_{0}(\theta_i)\big)$ in this case. Then, the $L^r$ convergence of $L_t^{(\varepsilon)}{1}_{E_2}$ follows from uniform integrability by (4.24).

2. (b) By (4.5), (4.20), and the monotone convergence theorem, we have, for $\boldsymbol{\theta}\in \Theta_*^p$ ,

   \begin{align*} &\mathbb{E}_{\mathcal{N}} L_t^*(\cdot\,,\boldsymbol{\theta})^r \\ &\quad= \frac{r!}{\Gamma(\beta_p)^r}\!\int_{0<x_1<\cdots<x_r<t} \Bigg(\!\prod_{i=1}^p (x_1-V_0(\theta_i))_+^{\beta-1} \!\Bigg) (x_2-x_1)^{\beta_p-1} \cdots (x_r-x_{r-1})^{\beta_p-1} \, {\rm d}\textbf{{x}}.\end{align*}
   Hence, by Fubini and (4.1),
   \begin{align*}\mathbb{E}^{\prime} & [(L_t^*)^r] \\ & = \frac{r!(1-\beta)^p}{\Gamma(\beta_p)^r} \int_{0<x_1<\cdots<x_r<t} \int_{[0,1]^p} \prod_{i=1}^p v^{-\beta}_i \Bigg(\prod_{i=1}^p (x_1-v_i)_+^{\beta-1} \Bigg) \, {\rm d}\textbf{{v}} \\ & \qquad\qquad\qquad \times (x_2-x_1)^{\beta_p-1} \cdots (x_r-x_{r-1})^{\beta_p-1} \, {\rm d}\textbf{{x}} \\& = \frac{r!(1-\beta)^p\mathrm{B}(1-\beta,\beta)^p}{\Gamma(\beta_p)^r} \int_{0<x_1<\cdots<x_r<t} (x_2-x_1)^{\beta_p-1} \cdots (x_r-x_{r-1})^{\beta_p-1} \, {\rm d}\textbf{{x}}.\end{align*}
   The last expression is equal to that in (4.18) through repeated applications of (4.1).

3. (c) This can be proved as [Reference Bai, Owada and Wang1, Lemma 2.7], so we only provide a sketch. Write

   \begin{equation*}\bigcap_{i=1}^p \left( R_0(h_i)+V_0(\theta_i)\right) =R^*(\textbf{{h}},\boldsymbol{\theta})+V^*(\textbf{{h}},\boldsymbol{\theta}),\end{equation*}
   where $V^*(\textbf{{h}},\boldsymbol{\theta})=\inf\cap_{i=1}^p \left(R_0(h_i)+V_0(\theta_i)\right)$ and $R^*$ is a $\beta_p$ -stable regenerative set starting at the origin independent of $V^*$ under $\mathbb{P}_V^p\times \mathbb{P}_\mathcal{N}^p$ [Reference Samorodnitsky and Wang25, Lemma 3.1]. By its construction, $(L_t^*)_{t\ge 0}$ is an additive functional which only increases on $R^*+V^* $ . In addition, by (4.18), increment-stationarity of $L_t^*$ , and Kolmogorov’s continuity theorem, $(L_t^*)_{t\ge 0}$ admits a $\mathbb{P}_\mathcal{N}^p\times \mathbb{P}_V^p$ -version which is continuous in t. Therefore, in view of [Reference Kingman14, Theorem 3] and [Reference Maisonneuve17, Theorem 3.1], the equality in (4.19) holds up to a positive multiplicative constant. This constant can be shown to be 1 by comparing the moments as in the proof of [Reference Bai, Owada and Wang1, Lemma 2.7]. □

4.2. Proof of Theorem 3.1

Proof. We first show the equivalence between the representations in Theorem 3.1(a) and (b), for which we shall fix $T>0$ and consider $t\in [0,T]$ . In view of Proposition 2.1 and Lemma 2.1, the representation in (3.2) is equivalent to

(4.25) \begin{equation}c_{p,\beta}\int_{(\textbf{{F}}\times [0,\infty))^p}^{\prime} L_t \Bigg(\bigcap_{i=1}^p (R_i +v_i)\Bigg) W^*({\rm d}R_1,{\rm d} v_1)\cdots W^*({\rm d} R_p,{\rm d} v_p),\end{equation}

where $W^*$ is a Gaussian random measure on $\textbf{{F}}\times [0,\infty)$ with control measure $\mathbb{P}_R\times \pi_V$ . Observe that, since $t\le T$ , the integrand above is zero if $v_i>T$ for some $i=1,\ldots,p$ . So, the integral domain $(\textbf{{F}}\times [0,\infty))^p$ in (4.25) can be replaced by $(\textbf{{F}}\times [0,T])^p$ , and $\pi_V$ can be viewed as its restriction on [0, T]. Then, define a Gaussian random measure by

(4.26) \begin{equation}W_T^*(\cdot)=\pi_V([0,T])^{-1/2} W^*(\cdot)=T^{(\beta-1)/2} W^*(\cdot),\end{equation}

whose control measure is now the probability measure $\mathbb{P}_R\times \widetilde{\pi}_V$ , where $\widetilde{\pi}({\rm d} v)=T^{\beta-1}(1-\beta) v^{-\beta}\,{\rm d} v$ , $v\in (0,T)$ . Substituting $W^*$ by $W_T^*$ in (4.25), the equivalence to the representation in (b) then follows from Lemma 2.1. Also, because of (4.26), the relation $c_{p,\beta,T}=T^{p(1-\beta)/2}c_{p,\beta}$ holds.

Next we prove (b). We shall assume $T=1$ for simplicity, and the argument is similar for general T. By Lemma 4.2, the $L^2$ linear isometry of multiple Wiener–Itô integrals (Section 2.1), and the standardization of variance at $t=1$ , the second moment of the expression in (4.25) when $t=1$ is equal to

\begin{equation*}p! \frac{2! \Gamma(\beta)^p \Gamma(2-\beta)^p}{\Gamma(\beta_p)\Gamma(\beta_p+2)} c_{p,\beta}^2=1.\end{equation*}

This implies (3.3).

Let $(H,\mathcal{H},\mathbb{P}_\mathcal{N})$ , $(\Theta,\mathcal{G},\mathbb{P}_V)$ , $L_t^*$ , $L_t^{(\varepsilon)}$ , $V_\varepsilon$ , and $R_\varepsilon$ be as in Lemma 4.2. Let $\nu$ and $U_\varepsilon$ be as described in the paragraph above (4.9). In view of Lemma 2.1, without loss of generality, one can set the probability space in (3.4) as $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})= (H\times \Theta, \mathcal{H}^{p}\times\mathcal{G}^p, \mathbb{P}_\mathcal{N}\times \mathbb{P}_{V})$ , assume that $ \mathbb{P}_{\mathcal{N}}\times \mathbb{P}_{V}$ is atomless, and choose $R=R_0$ and $V=V_0$ . In view of (4.19), for $0 \le t\le 1$ we have, almost surely (a.s.),

\begin{align*}\widetilde{Z}(t)\,:\!=c_{p,\beta}^{-1}Z(t) = \int_{(H\times\Theta )^p}^{\prime} L_t^*(\textbf{{h}},\boldsymbol{\theta}) W({\rm d} h_1,{\rm d}\theta_{1})\cdots W({\rm d} h_p,{\rm d}\theta_{p}).\end{align*}

For $\varepsilon>0$ , define

(4.27) \begin{equation}\widetilde{Z}_{\varepsilon}(t)=\int_{(H\times\Theta )^p}^{\prime} L_t^{(\varepsilon)}(\textbf{{h}},\boldsymbol{\theta}) W({\rm d} h_1,{\rm d}\theta_{1})\cdots W({\rm d} h_p,{\rm d}\theta_{p}).\end{equation}

By Lemma 4.2, $L_t^{(\varepsilon)}$ converges to $L_t$ in $L^2$ as $\varepsilon=1/n\rightarrow 0$ . So, by the $L^2$ linear isometry of multiple integrals, as $\varepsilon=1/n\rightarrow 0$ ,

(4.28) \begin{equation}\widetilde{Z}_{\varepsilon}(t)\overset{L^2(\Omega)}{\longrightarrow} \widetilde{Z}(t), \qquad t\ge 0.\end{equation}

Next, for $\varepsilon>0$ , define the Gaussian processes

\begin{align*} G_{\varepsilon}(x) & = \left(\frac{\pi_\varepsilon([0,1])}{ d_\varepsilon}\right)^{1/2} \int_{H\times \Theta} {1}_{\{x\in R_\varepsilon(h)+V_{\varepsilon}(\theta)\}} W({\rm d} h,{\rm d}\theta), \qquad x\in [0,1], \\[3pt] G_{\varepsilon}^*(x) & = d_\varepsilon^{-1/2} \int_{H\times \Theta} {1}_{\{x\in R_\varepsilon(h)+U_{\varepsilon}(\theta)\}} W({\rm d} h,{\rm d}\theta), \qquad x\in [0,\infty).\end{align*}

We claim that the Gaussian process $(G_{\varepsilon}^*(x))_{x\in [0,\infty)}$ is stationary and

(4.29) \begin{equation}(G_\varepsilon(x))_{x\in [0,1]}\overset{{\rm f.d.d.}}{=}(G_\varepsilon^*(x))_{x\in [0,1]},\end{equation}

where $\overset{{\rm f.d.d.}}{=}$ means equality in finite-dimensional distributions. Indeed, for $0\le x\le y$ , in view of (4.9),

\begin{align*}\mathbb{E} [G_\varepsilon^*(x)G_\varepsilon^*(y)] & = d_\varepsilon^{-1} \nu(\{x,y\}\subset R_\varepsilon+U_\varepsilon) = d_\varepsilon^{-1} \nu(\{0,y-x\}\subset R_\varepsilon+U_\varepsilon ) \\[3pt] & =\mathbb{E} G_\varepsilon^*(0)G_\varepsilon^*(y-x).\end{align*}

On the other hand, for $0\le x\le y\le 1$ ,

\begin{align*}\mathbb{E}[ G_\varepsilon(x)G_\varepsilon(y)] &= (\pi_\varepsilon([0,1])/d_\varepsilon) \mathbb{P}^{\prime}(\{x,y\}\subset R_\varepsilon+V_\varepsilon)= d_\varepsilon^{-1} \nu(\{x,y\}\subset R_\varepsilon+U_\varepsilon, U_\varepsilon \le 1)\\[3pt] & = d_\varepsilon^{-1} \nu(\{x,y\}\subset R_\varepsilon+U_\varepsilon)=\mathbb{E} G_\varepsilon^*(x)G_\varepsilon^*(y).\end{align*}

The reason for introducing $G^*_\varepsilon$ in addition to $G_\varepsilon$ is because we will apply the spectral representation, see (4.30), of a stationary Gaussian process defined on $\mathbb{R}$ . While $G^*_\varepsilon$ defined on $[0,\infty)$ can be extended to a stationary Gaussian process on $\mathbb{R}$ by shift, it is not the case for $G_\varepsilon$ defined only on [0, 1].

Next, because $0\in R_\varepsilon$ , we have $\{0,x\}\in R_\varepsilon+U_\varepsilon$ if and only if $U_\varepsilon=0$ and $x\in R_\varepsilon$ . So, by (4.7) and (4.8), $\mathbb{E} [G_\varepsilon^*(0) G_\varepsilon^*(x)] =d_\varepsilon^{-1}\pi_\varepsilon(\{0\}) p_\varepsilon(x) =p_\varepsilon(x)$ , and, in particular, $\mathbb{E} G_\varepsilon^*(0)^2=p_\varepsilon(0)=1$ . Using the spectral representation in [Reference Janson12, Theorem 7.54], we have, for $\varepsilon>0$ and $x\ge 0$ , that

(4.30) \begin{equation}(G_\varepsilon^*(x))_{x\in [0,\infty)}\overset{{\rm f.d.d.}}{=}\left(\int_{\mathbb{R}} {\rm e}^{{\rm i}\lambda x} \widehat{W}_\varepsilon({\rm d}\lambda)\right)_{x\in [0,\infty)},\end{equation}

where $\widehat{W}_\varepsilon$ is a complex-valued Hermitian Gaussian random measure with control measure $\mu_\varepsilon$ satisfying

(4.31) \begin{equation}p_{\varepsilon}(x)=\int_\mathbb{R} {\rm e}^{{\rm i}\lambda x} \mu_\varepsilon({\rm d}\lambda),\qquad x\ge 0.\end{equation}

In addition, using Gaussian moments, we have, for some constant $c>0$ not depending on x, that

\begin{equation*}\mathbb{E}[G^*(x)-G^*(0)]^4=3(\mathbb{E}[G^*(x)-G^*(0)]^2)^2= 12(1-p_{\varepsilon}(x))^2\le c x^2,\qquad x\ge 0,\end{equation*}

where the inequality follows from an examination of (4.3). Hence, $G_\varepsilon$ and $G_\varepsilon^*$ admit continuous versions by Kolmogorov’s continuity theorem. We shall work with such versions when integrating along their time variables below.

Applying a stochastic Fubini theorem [Reference Peccati and Taqqu21, Theorem 5.13.1] to (4.27), see also (4.17), and using the relation between Hermite polynomials and multiple Wiener–Itô integrals [Reference Janson12, Theorem 7.52, Equation (7.23), and Theorem 3.19], (4.29), and (4.30), we have

\begin{align*} \widetilde{Z}_{\varepsilon}(t)& \ \overset{\text{a.s.}}{=}\frac{1}{\Gamma(\beta_p)} \left( \frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \left(\frac{d_\varepsilon}{ \pi_\varepsilon([0,1]) } \right)^{p/2} \int_0^t H_p\left(G_{\varepsilon}(x)\right) {\rm d} x \notag\\[3pt] & \overset{{\rm f.d.d.}}{=} \frac{1}{\Gamma(\beta_p)} \left( \frac{\varepsilon}{{\rm e}}\right)^{\beta_p-1} \left(\frac{d_\varepsilon}{ \pi_\varepsilon([0,1]) } \right)^{p/2} \int_0^t H_p\left(G_{\varepsilon}^*(x)\right) {\rm d} x \notag\\[3pt] & \overset{{\rm f.d.d.}}{=} \frac{\big(d_\varepsilon \left(\varepsilon/{\rm e}\right)^{\beta-1}\big)^p }{\Gamma(\beta_p)\pi_\varepsilon([0,1])^{p/2}} \int ^{\prime\prime}_{\mathbb{R}^p} \frac{{\rm e}^{{\rm i}\big(\sum_{j=1}^p \lambda_j\big)t}-1}{{\rm i}\big(\sum_{j=1}^p \lambda_j\big)} \widetilde{W}_{\varepsilon}({\rm d}\lambda_1)\cdots \widetilde{W}_\varepsilon({\rm d}\lambda_p),\end{align*}

where $\widetilde{W}_\varepsilon\,:\!=d_\varepsilon^{-1/2}\widehat{W}_\varepsilon$ has control measure $\widetilde{\mu}_\varepsilon\,:\!=d_\varepsilon^{-1}\mu_\varepsilon$ . By (4.31) and (4.4),

(4.32) \begin{equation}\int_{\mathbb{R}} {\rm e}^{{\rm i}\lambda x} \widetilde{\mu}_\varepsilon({\rm d}\lambda)=d_\varepsilon^{-1}p_{\varepsilon}(|x|)=u_\varepsilon(|x|).\end{equation}

Note that, in view of (4.5) and (4.6), as $\varepsilon\rightarrow 0$ ,

(4.33) \begin{equation} u_\varepsilon(x)\rightarrow u_0(x)= \frac{ x^{\beta-1}}{\Gamma(\beta)}, \qquad x> 0. \end{equation}

Define

\begin{equation*}\widetilde {\mu}_0({\rm d}\lambda)\,:\!=c_\beta |\lambda|^{-\beta} \, {\rm d}\lambda, \qquad \lambda\neq 0,\end{equation*}

where $c_\beta=\frac{2(1-\beta)}{\Gamma(\beta)}\int_0^\infty \sin(y)y^{\beta-2}\,{\rm d} y$ is a constant ensuring the relation

(4.34) \begin{equation}4\int_0^\infty \frac{\sin(ax)}{x} u_0(x)\,{\rm d} x = \widetilde{\mu}_0([-a,a]), \qquad a>0,\end{equation}

which can be obtained via a change of variable $ax=y$ in the integral above.

We claim that, as $\varepsilon\rightarrow 0$ ,

(4.35) \begin{align} (\widetilde{Z}_\varepsilon(t))_{0 \le t\le 1}\!\overset{{\rm f.d.d.}}{\longrightarrow} \! \Bigg(\frac{\Gamma(\beta)^p }{\Gamma(\beta_p) \Gamma(2-\beta)^{p/2}}\int ^{\prime\prime}_{\mathbb{R}^p} \frac{{\rm e}^{{\rm i}\big(\sum_{j=1}^p \lambda_j\big)t}-1}{{\rm i}\big(\sum_{j=1}^p \lambda_j\big)} \widehat{W}_0({\rm d}\lambda_1)\cdots \widehat{W}_0({\rm d}\lambda_p)\Bigg)_{_{0 \le t\le 1}},\end{align}

where $\overset{{\rm f.d.d.}}{\longrightarrow}$ stands for the convergence of the finite-dimensional distributions and $\widehat{W}_\varepsilon$ is a complex-valued Hermitian Gaussian random measure with control measure $\widetilde{\mu}_0$ . The right-hand side of (4.35) is, up to a constant, the frequency-domain representation of a Hermite process (1.3) after noticing that $ \widehat{W}_0({\rm d}\lambda) \overset{{\rm d}}{=} c_{\beta}^{-1/2} |\lambda|^{-\beta/2}\widehat{W}({\rm d}\lambda)$ . If (4.35) holds, then in view of (4.28), the proof is concluded.

To show (4.35), in view of (4.6), (4.13), the Cramér–Wold device, and [Reference Dobrushin and Major7, Lemma 3] (see also [Reference Pipiras and Taqqu23, Proposition 5.3.6]), it suffices to show, as $\varepsilon\rightarrow 0$ , the vague convergence

(4.36) \begin{equation}\widetilde{\mu}_\varepsilon({\rm d}\lambda)\overset{{\rm v}}{\rightarrow} \widetilde {\mu}_0({\rm d}\lambda),\end{equation}

as well as

(4.37) \begin{equation}\lim_{A\rightarrow\infty}\limsup_{\varepsilon\rightarrow 0}\int_{([-A,A]^p)^{\rm c}} |k_{t}(\lambda_1,\ldots,\lambda_p)|^2 \widetilde{\mu}_\varepsilon({\rm d}\lambda_1)\cdots \widetilde{\mu}_\varepsilon({\rm d}\lambda_p)=0,\end{equation}

where

\begin{equation*}k_t(\lambda_1,\ldots,\lambda_p)\,:\!=\int_{0}^t {\rm e}^{{\rm i} s(\lambda_1+\cdots+\lambda_p)} \, {\rm d} s = \frac{\exp\big({\rm i}\big(\sum_{j=1}^p \lambda_j\big)t\big)-1}{{\rm i}\big(\sum_{j=1}^p \lambda_j\big)}, \qquad t> 0, \ \sum_{j=1}^p \lambda_j\neq 0.\end{equation*}

We first show (4.36). By an inversion of the Fourier transform (4.32) [Reference Lindgren16, Theorem 4:4] we have, for any $a>0$ ,

\begin{align*}\widetilde{\mu}_\varepsilon([-a,a])= \lim_{A\rightarrow\infty}\int_{-A}^A \frac{{\rm e}^{-{\rm i} ax}-{\rm e}^{{\rm i} ax}}{-{\rm i} x} u_\varepsilon(x)\,{\rm d} x = 4\int_{0}^\infty \frac{\sin(ax)}{x} u_\varepsilon(x)\,{\rm d} x,\end{align*}

where, for the last expression, its integrability follows from the fact (see (4.4), (4.6), and (4.22)) that

(4.38) \begin{equation}u_\varepsilon(x)\le c x^{\beta-1},\qquad x>0.\end{equation}

Note that in the inversion above, we have implicitly used the continuity of $\widetilde{\mu}_\varepsilon([-a,a])$ in a, which can be verified using the dominated convergence theorem via the bound $\sup_{a\in [0,b]}|\sin(ax)|\le (bx)\wedge 1$ for any $x,b>0$ . Then, by (4.38), (4.33), (4.34), and the dominated convergence theorem, we conclude that, as $\varepsilon\rightarrow 0$ ,

\begin{equation*}\widetilde{\mu}_\varepsilon([-a,a])\rightarrow \widetilde{\mu}_0([-a,a]),\end{equation*}

and thus (4.36) holds.

To show (4.37), define a measure on $\mathbb{R}^p$ as

\begin{equation*}\kappa_\varepsilon({\rm d}\lambda_1,\ldots,{\rm d}\lambda_p)\,:\!=|k_t(\lambda_1,\ldots,\lambda_p)|^2 \widetilde{\mu}_\varepsilon({\rm d}\lambda_1)\cdots \widetilde{\mu}_\varepsilon({\rm d}\lambda_p).\end{equation*}

We shall obtain (4.37) as a tightness condition from the weak convergence of $\kappa_\varepsilon$ as $\varepsilon\rightarrow 0$ . Indeed, set $\textbf{1}=(1,\ldots,1)\in \mathbb{R}^p$ and let $\langle \cdot\, , \cdot \rangle$ denote the Euclidean inner product. By (4.32), (4.38), Fubini, and the dominated convergence theorem, we have, as $\varepsilon\rightarrow 0$ ,

\begin{align*}\int_{\mathbb{R}^p} {\rm e}^{{\rm i} \langle \boldsymbol{\lambda} ,\textbf{{x}} \rangle } \kappa_\varepsilon({\rm d}\boldsymbol{\lambda})&= \int_{\mathbb{R}^p} \widetilde{\mu}_\varepsilon^p({\rm d}\boldsymbol{\lambda}) {\rm e}^{{\rm i} \langle \boldsymbol{\lambda} , \textbf{{x}}\rangle } \int_{0}^t {\rm e}^{{\rm i} s_1 \langle \boldsymbol{\lambda},\textbf{1}\rangle} \,{\rm d} s_1\int_{0}^t {\rm e}^{-{\rm i} s_2 \langle\boldsymbol{\lambda}, \textbf{1}\rangle} \, {\rm d} s_2 \\[3pt] & = \int_0^t{\rm d} s_1\int_0^t{\rm d} s_2 \prod_{j=1}^p u_\varepsilon(|x_j +s_1-s_2|) \\&\rightarrow \int_0^t{\rm d} s_1\int_0^t{\rm d} s_2 \prod_{j=1}^p u_0(|x_j +s_1-s_2|)\\ &=\frac{t^{\beta_p+1}}{\Gamma(\beta)^{p}}\int_{-1}^{1} (1-|y|)\prod_{j=1}^p |x_j/t+y|^{\beta-1}\,{\rm d} y =\!:\, \phi(\textbf{{x}}),\end{align*}

where the last line is obtained by the change of variables $s_1=t(y+w)$ , $s_2=t w$ and integrating out the variable w. Note that $\phi(\textbf{{x}})<\infty$ for any $\textbf{{x}}\in \mathbb{R}^p$ since $(\beta-1)p>-1$ . Furthermore, the function $\phi(\textbf{{x}})$ is continuous [Reference Dobrushin and Major7, Lemma 1]. So, tightness (4.37) holds by Lévy’s continuity theorem. Hence, (4.35) is established and thus the proof is complete. □