# Scaling limits of nonlinear functions of random grain model,
with application to Burgers’ equation

Donatas Surgailis

(
Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24,
03225 Vilnius, Lithuania
)

###### Abstract

We study scaling limits of nonlinear functions $G$ of random grain model $X$
on $\mathbb{R}^{d}$ with long-range dependence and marginal Poisson
distribution. Following [15] we assume that the intensity $M$ of the
underlying Poisson process of grains increases together with the scaling
parameter $\lambda$ as $M=\lambda^{\gamma}$, for some $\gamma>0$. The results
are applicable to the Boolean model and exponential $G$ and rely on an
expansion of $G$ in Charlier polynomials and a generalization of Mehler’s
formula. Application to solution of Burgers’ equation with initial aggregated
random grain data is discussed.

Keywords: Random grain model, nonlinear function, Boolean model, long-range
dependence, scaling limit, aggregation, Poisson distribution, Charlier
polynomials, Mehler’s formula, Gaussian random field, stable random field,
Burgers’ equation, initial random grain data

##  1 Introduction

Limit theorems for random fields (RFs) with continuous or discrete
$d$-dimensional argument have been extensively studied in the literature.
Quite often, such work refer to the limit distribution of integrals

$\displaystyle X_{\lambda}(\phi)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{d}}X(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t},\qquad\text{as}\
\lambda\to\infty,$ (1.1)

(or respective sums in the discrete argument case), where
$X=\\{X(\boldsymbol{t});\boldsymbol{t}\in\mathbb{R}^{d}\\}$ is a given
stationary RF, for each $\phi$ from a class of (test) functions
$\Phi=\\{\phi:\mathbb{R}^{d}\to\mathbb{R}\\}$. A suitably normalized limit of
(1.1) is a RF indexed by $\phi\in\Phi$, called the (isotropic) scaling limit
of $X$ in this paper. The above approach is common in the theory of
generalized RFs, and is discussed in [5] together with various classes of
$\Phi$. For $\phi(\boldsymbol{t})=\mathbb{I}(\boldsymbol{t}\in A)$, where $A$
is a bounded Borel subset of $\mathbb{R}^{d}$, (1.1) is the integral (sum) of
$X(\boldsymbol{t})$’s over $\boldsymbol{t}\in\lambda A$ whose limit
distribution was studied in [18] for linear RF $X$ on $\mathbb{Z}^{d}$ and $A$
having irregular boundary. For $\Phi=\Phi_{{\rm
rec},d}:=\\{\mathbb{I}(\boldsymbol{t}\in]\boldsymbol{0},\boldsymbol{x}]);\boldsymbol{x}\in\mathbb{R}^{d}_{+}\\},$
$]\boldsymbol{0},\boldsymbol{x}]:=\prod_{i=1}^{d}]0,x_{i}]$, (1.1) present a
$d$-dimensional analog of the partial integral process of time series, leading
to a limit RF indexed by $\boldsymbol{x}\in\mathbb{R}^{d}_{+}$.

A stationary RF $X$ on $\mathbb{R}^{d}$ with finite variance with is said
long-range dependent (LRD) if its covariance function
$r_{X}(\boldsymbol{t}):={\rm Cov}(X(\boldsymbol{0}),X(\boldsymbol{t}))$ is not
integrable:
$\int_{\mathbb{R}^{d}}|r_{X}(\boldsymbol{t})|\mathrm{d}\boldsymbol{t}=\infty$.
It is well-known that LRD RFs can display a variety of Gaussian and non-
Gaussian limit behaviors and scaling limits, see [6, 20, 36, 31] and the
references therein. The framework in (1.1) can be modified by replacing the
isotropic scaling $\boldsymbol{t}\to\boldsymbol{t}/\lambda$ in the test
function with anisotropic scaling
$\boldsymbol{t}\to\lambda^{-\Gamma}\boldsymbol{t},$ where $\Gamma={\rm
diag}(\gamma_{1},\cdots,\gamma_{d}),(\gamma_{1},\cdots,\gamma_{d})=\boldsymbol{\gamma}\in\mathbb{R}^{d}_{+}$.
It was observed in [34, 35, 27, 28, 41, 42] that in dimensions $d=2,3$ and a
rich class of LRD RFs (including Gaussian and linear RFs) the corresponding
anisotropic limits in (1.1) exist for any
$\boldsymbol{\gamma}\in\mathbb{R}^{d}_{+},\phi\in\Phi_{{\rm rec},d}$ and
depend on $\boldsymbol{\gamma}$. Particularly, when $d=2$ a scaling transition
may occur meaning that there exists a critical $\gamma^{0}>0$ depending on
internal parameters of RF $X$ such that the limit RFs do not depend on
$\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2})$ for
$\gamma_{2}/\gamma_{1}>\gamma^{0}$ and $\gamma_{2}/\gamma_{1}<\gamma^{0}$ and
exhibit a trichotomy of the (anisotropic) scaling behavior [34, 35, 28, 42].

The above scaling procedures can be extended to include aggregation, as
follows. Let $X_{i},i=1,2,\cdots,M$ be independent copies of RF $X$ in (1.1).
Consider the limit distribution of the ‘aggregate’

$X_{\lambda,M}(\phi)\ :=\
\sum_{i=1}^{M}\int_{\mathbb{R}^{d}}X_{i}(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t}$
(1.2)

as $\lambda\to\infty$ and $M$ increases with $\lambda$ at a certain rate. In
as follows, we refer to the limit distribution of the sums in (1.2) as the
scaling limit with aggregation. By concretising $M$ as $M=[\lambda^{\gamma}y]$
for some $\gamma>0,y>0$ we see that (1.2) may be regarded as a discretized
version of the integral in (1.1) for stationary RF $X^{\prime}$ on
$\mathbb{R}^{d}\times\mathbb{Z}$ given by
$X^{\prime}(\boldsymbol{t},s):=X_{s}(\boldsymbol{t}),(\boldsymbol{t},s)\in\mathbb{R}^{d}\times\mathbb{Z}$,
corresponding to (partly) anisotropic scaling
$(\boldsymbol{t},s)\to(\boldsymbol{t}/\lambda,s/\lambda^{\gamma})$ and test
function
$\phi^{\prime}(\boldsymbol{t},s):=\phi(\boldsymbol{t})\mathbb{I}(s\in]0,y])$.
Scaling limits with aggregation in dimension $d=1$ were studied in [11, 16,
19, 22, 23, 26, 25, 30] and other works, in connection with applications in
communications and econometrics. As noted in [19], the trichotomy of the limit
distribution observed in these papers can be interpreted as a scaling
transition for RFs on the plane. Iterated limits of (1.2) when $M\to\infty$
(first) and $\lambda\to\infty$ (second) and/or vice versa, were discussed in
[2, 33, 35] and other above cited references. A particularly simple form of
aggregation occurs in models based on Poisson process as in (1.3) where
summing over $M$ independent copies reduces to multiplying by $M$ the
intensity of the underlying Poisson process [3, 4, 15, 16, 19, 22].

Unsurprisingly, most existing results on scaling limits of LRD RFs ($d\geq 2$)
(with or without aggregation) apply to linear models. A notable exception is
Gaussian subordinated RFs (written as a nonlinear function
$G(X(\boldsymbol{t}))$ of a Gaussian LRD RF $X$), treated via Hermite
expansion in the classical work [6]. For more recent developments on chaos
expansions and limit theorems under Gaussian subordination, see [24]. This is
in contrast to the one-dimensional case $d=1$, where the martingale approach
developed in [14] is applicable to nonlinear functions and statistics of LRD
moving averages. See monographs [8, 13]. Scaling limits of polynomial
functions $G$ of LRD moving-average RFs $X$ are discussed in [38, 28]. The
case of indicator functions $G(x)=\mathbb{I}(x\leq y)$ for the same class of
RFs was considered in [7]. See [17] for statistical application.

A class of LRD RFs whose trajectories and distribution is very different from
Gaussian and moving-average RFs are random grain (RG) models, defined as
follows. Let $\\{\boldsymbol{u}_{j};j\geq 1\\}\subset\mathbb{R}^{d}$ be a
Poisson point process with intensity $\mathrm{d}\boldsymbol{u}$. Let be given
a bounded Borel set $\Xi^{0}\subset\mathbb{R}^{d}$ (‘generic grain’) and and
an i.i.d. sequence $R,R_{j};j\geq 1$ of r.v.s with values in $\mathbb{R}_{+}$,
distribution $F(\mathrm{d}r)$, finite expectation
$\mathrm{E}R=\int_{\mathbb{R}_{+}}rF(\mathrm{d}r)<\infty$, and independent of
the Poisson process. The RG RF is obtained by counting at each point
$\boldsymbol{t}$ the number of randomly dilated grains
$\boldsymbol{u}_{j}+R_{j}^{1/d}\Xi^{0}$ ‘centered’ at $\boldsymbol{u}_{j}$
which cover it, viz.,

$X(\boldsymbol{t}):=\sum_{j=1}^{\infty}\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}_{j}\in
R_{j}^{1/d}\Xi^{0}),\qquad\boldsymbol{t}\in\mathbb{R}^{d}.$ (1.3)

For $d=1$ and $\Xi^{0}=]0,1]$, (1.3) is the number of customers in a
stationary M/G/$\infty$ queue with service time distribution $F(\mathrm{d}r)$.
The RF in (1.3) is infinitely divisible and has a Poisson stochastic integral
representation

$X(\boldsymbol{t})=\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0}){\cal
N}(\mathrm{d}\boldsymbol{u},\mathrm{d}r),\qquad\boldsymbol{t}\in\mathbb{R}^{d},$
(1.4)

where ${\cal N}(\mathrm{d}\boldsymbol{u},\mathrm{d}r)$ is a Poisson random
measure with mean $\mathrm{E}{\cal
N}(\mathrm{d}\boldsymbol{u},\mathrm{d}r)=\mathrm{d}\boldsymbol{u}F(\mathrm{d}r)$.
RF in (1.4) is stationary and has marginal Poisson distribution with mean
$\mu:=\mathrm{E}X(\boldsymbol{t})=\int_{\mathbb{R}^{d}}\mathrm{P}(\boldsymbol{t}-\boldsymbol{u}\in
R^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{u}={\rm Leb}_{d}(\Xi^{0})\mathrm{E}R$. It
is well-known [15] that under mild conditions on $\Xi^{0}$ the RG model is LRD
if the distribution $F$ varies regularly at infinity with exponent
$\alpha\in(1,2)$. The conditions on $F$ and $\Xi^{0}$ in this paper
(Assumption LRD in Sec.2) imply that

$\displaystyle r_{X}(\boldsymbol{t})$ $\displaystyle\sim$
$\displaystyle\|\boldsymbol{t}\|^{-d(\alpha-1)}\ell(\frac{\boldsymbol{t}}{\|\boldsymbol{t}\|}),\qquad|\boldsymbol{t}|\to\infty,\quad
1<\alpha<2$

where $\ell(\boldsymbol{z}),\|\boldsymbol{z}\|=1$ is a continuous (angular)
function on the unit sphere of $\mathbb{R}^{d}$ given in (2.3) and
$\int_{\mathbb{R}^{d}}|r_{X}(\boldsymbol{t})|\mathrm{d}\boldsymbol{t}=\infty$
holds for any $d\geq 1,\alpha\in(1,2)$.

Scaling limits with aggregation of RG model in (1.4) were discussed in [15]
(see also [3, 4]). Let

$X_{M}(\boldsymbol{t}):=\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\,\Xi^{0}){\cal
N}_{M}(\mathrm{d}\boldsymbol{u},\mathrm{d}r),\qquad\boldsymbol{t}\in\mathbb{R}^{d}$
(1.5)

be random grain model with Poisson intensity
$M\mathrm{d}\boldsymbol{u}F(\mathrm{d}r)$, a multiple of the intensity in
(1.4), where $M>0$. [15] proved that when $M$ increases with $\lambda$ at
certain rate (e.g., $M=\lambda^{\gamma}$ for some $\gamma>0$), scaling limits
of (1.5) exhibit a trichotomy depending on $\gamma$ (described in sec. 2).

The main object of this paper are scaling limits of subordinated RFs

$\displaystyle Y(\boldsymbol{t})=G(X(\boldsymbol{t}))\quad\text{and}\quad
Y_{M}(\boldsymbol{t})$ $\displaystyle:=$ $\displaystyle
G\big{(}\frac{X_{M}(\boldsymbol{t})-\mathrm{E}X_{M}(\boldsymbol{t})}{M^{1/2}}\big{)},\quad\boldsymbol{t}\in\mathbb{R}^{d},$
(1.6)

where $X(\boldsymbol{t}),X_{M}(\boldsymbol{t})$ are as in (1.4), (1.5) and
$G(x)$ is a nonlinear function with
$\mathrm{E}G(X(\boldsymbol{0}))^{2}<\infty$ and
$\mathrm{E}Y^{2}_{M}(\boldsymbol{0})=\mathrm{E}G^{2}_{M}(X_{M}(\boldsymbol{0}))<\infty,G_{M}(x):=G((x-M\mu)/M^{1/2})$.
In other words, we discuss distributional limits of normalized integrals

$Y_{\lambda}(\phi)=\int_{\mathbb{R}^{d}}Y(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t}\quad\text{and}\quad
Y_{\lambda,M}(\phi)=\int_{\mathbb{R}^{d}}Y_{M}(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t}$
(1.7)

for a large class of test functions $\phi$, as $\lambda\to\infty$ and
$M=\lambda^{\gamma}$ for some $\gamma>0$. Our main result - Theorem 2 \- says
that the scaling limits of (1.6) and (1.5) are essentially the same (including
the trichotomy of the limit distribution) provided the Hermite rank of $G$ is
1, or

$h_{G,\mu}(1):=\mathrm{E}[G(Z_{\mu})Z_{\mu}]\neq 0,\qquad Z_{\mu}\sim
N(0,\mu),$ (1.8)

in which case the difference between the scaling limits of (1.6) and (1.5)
reduces to the multiplicative factor in (1.8). We also prove a similar result
for scaling limits of $Y(\boldsymbol{t})=G(X(\boldsymbol{t}))$ with (1.8)
replaced by the first coefficient of the expansion of $G$ in Charlier
polynomials (Proposition 2).

The proofs of our results are rather simple, relying on expansion of bivariate
Poisson distribution in orthogonal Charlier polynomials and a Poissonian
analog of Mehler’s formula (Lemma 1). We hope that this approach can be useful
to other Poisson-based RF models and nonlinear triangular arrays. Some open
problems are indicated in Remarks 5, 6, 7, and 8.

In this paper, the limit results are applied to two subordinated RG models.
The first one is $G(x)=x\wedge 1,\ x\in\mathbb{N}$, or
$\hat{X}(\boldsymbol{t}):=X(\boldsymbol{t})\wedge
1,\boldsymbol{t}\in\mathbb{R}^{d}$, referred to below as the Boolean model. By
taking $\phi(\boldsymbol{t})=\mathbb{I}(\boldsymbol{t}\in A)$, where
$A\subset\mathbb{R}^{d}$ is a bounded Borel set, we see that

$\int_{\mathbb{R}^{d}}\mathbb{I}(\boldsymbol{t}/\lambda\in
A)\hat{X}(\boldsymbol{t})\mathrm{d}\boldsymbol{t}={\rm Leb}_{d}({\cal
X}\cap\lambda A)=:\hat{X}_{\lambda}(A)$ (1.9)

is the ‘volume’ of the intersection of the Boolean set ${\cal
X}:=\bigcup_{j=1}^{\infty}(\boldsymbol{u}_{j}+R_{j}\Xi^{0})\subset\mathbb{R}^{d}$
with large ‘inflated’ set $\lambda A$. We remark that the Boolean model is a
basic model in stereology and stochastic geometry [37]. According to Corollary
2, under the above assumptions on (1.3), the limit distribution of (1.9) is
asymmetric $\alpha$-stable for a general set $A$. The second example is the
exponential function $G(x)=\mathrm{e}^{ax},x\in\mathbb{R}$, where $a\neq 0$ is
a real parameter. We have that $h_{G,\mu}(1)=a\mathrm{e}^{a^{2}\mu/2}$
satisfies (1.8) and Theorem 2 applies to the above $G$, see Example 1. The
case $G(x)=\mathrm{e}^{ax}$ is particular interest to the study of scaling
limits of statistical solution of Burgers’ equation with random linear data as
in (1.5) discussed in the last Section 4.

Notation. In this paper, $\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}$ (respectively,
$\begin{array}[t]{c}\stackrel{{\scriptstyle\rm fdd}}{{\longrightarrow}}\\\
\end{array}$) denote respectively the weak convergence of distributions
(respectively, finite dimensional distributions). $C$ stands for a generic
positive constant which may assume different values at various locations and
whose precise value has no importance.
$\mathbb{R}^{d}:=\\{\boldsymbol{t}=(t_{1},\cdots,t_{d}):t_{i}\in\mathbb{R},i=1,\cdots,d\\},\boldsymbol{0}=(0,\cdots,0)\in\mathbb{R}^{d},\mathbb{R}^{d}_{+}:=(0,\infty)^{d}$.
$\mathbb{I}(A)$ denotes the indicator function of a set $A$.

##  2 Scaling limits of RG model

Most results in this section either belong to [15], or are variations of the
latter work. For reader’s convenience, short complete proofs are included. The
following assumption is used throughout this paper without further notice.

Assumption LRD $\Xi^{0}\subset\mathbb{R}^{d}$ is a bounded Borel set whereas
$F(\mathrm{d}r)=f(r)\mathrm{d}r$ has a density function such that

$\displaystyle f(r)$ $\displaystyle\sim$ $\displaystyle
c_{f}r^{-1-\alpha},\quad r\to\infty\qquad(\exists\,c_{f}>0,\ \alpha\in(1,2)).$
(2.1)

Moreover, the function $(r,\boldsymbol{z})\mapsto{\rm
Leb}_{d}\big{(}\Xi^{0}\cap(\Xi^{0}-r^{-1/d}\,\boldsymbol{z})\big{)}$ is
continuous in
$(r,\boldsymbol{z})\in\mathbb{R}_{+}\times\\{\|\boldsymbol{z}\|=1\\}$ and
nontrivial.

The class of test functions in (1.1) and elsewhere in this paper is

$\Phi=L^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d}),$ (2.2)

where $L^{1}(\mathbb{R}^{d})$ ($L^{\infty}(\mathbb{R}^{d})$) stand for the
linear space of all Borel functions $\phi:\mathbb{R}^{d}\to\mathbb{R}$ such
that
$\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t})|\mathrm{d}\boldsymbol{t}<\infty$
(respectively, such that
$\sup_{\boldsymbol{t}\in\mathbb{R}^{d}}|\phi(\boldsymbol{t})|<\infty$). The
proofs of Theorems 1 and 2 use the fact that any Borel function is a.e.
continuous on $\mathbb{R}^{d}$ (Lusin’s theorem).

###### Proposition 1

(i) Relation (1) holds with bounded, continuous and nonnegative angular
function

$\displaystyle\ell(\boldsymbol{z})$ $\displaystyle:=$ $\displaystyle
c_{f}\int_{\mathbb{R}_{+}}{\rm
Leb}_{d}\big{(}\Xi^{0}\cap(\Xi^{0}-r^{-1/d}\,\boldsymbol{z})\big{)}r^{-\alpha}\mathrm{d}r.$
(2.3)

(ii) For any $\phi\in\Phi$ as $\lambda\to\infty$

$\displaystyle
c_{\lambda}(\phi):=\int_{\mathbb{R}^{2d}}\phi(\boldsymbol{t}_{1}/\lambda)\phi(\boldsymbol{t}_{2}/\lambda)r_{X}(\boldsymbol{t}_{1}-\boldsymbol{t}_{2})\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}$
$\displaystyle\sim$ $\displaystyle\lambda^{d(3-\alpha)}c(\phi),$ (2.4)

where

$c(\phi):=\int_{\mathbb{R}^{2d}}\phi(\boldsymbol{t}_{1})\phi(\boldsymbol{t}_{2})\ell\big{(}\frac{\boldsymbol{t}_{1}-\boldsymbol{t}_{2}}{\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|}\big{)}\frac{\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}}{\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|^{d(\alpha-1)}}$
(2.5)

and the integral on the r.h.s. of (2.5) converges. Moreover,

$\displaystyle\int_{\mathbb{R}^{2d}}\phi(\boldsymbol{t}_{1}/\lambda)\phi(\boldsymbol{t}_{2}/\lambda)(1\wedge
r^{2}_{X}(\boldsymbol{t}_{1}-\boldsymbol{t}_{2}))\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}$
$\displaystyle=$ $\displaystyle\begin{cases}O(\lambda^{d}),&\alpha>3/2,\\\
O(\lambda^{2d(2-\alpha)}),&\alpha<3/2,\\\
O(\lambda^{d}\log\lambda),&\alpha=3/2.\end{cases}$ (2.6)

Proof. (i) From (1.5) we have that

$\displaystyle\|\boldsymbol{t}\|^{d(\alpha-1)}r_{X}(\boldsymbol{t})$
$\displaystyle=$
$\displaystyle\|\boldsymbol{t}\|^{d(\alpha-1)}\int_{0}^{\infty}{\rm
Leb}_{d}(r^{1/d}\Xi^{0}\cap(r^{1/d}\Xi^{0}-\boldsymbol{t}))f(r)\mathrm{d}r$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\rm
Leb}_{d}\big{(}\Xi^{0}\cap(\Xi^{0}-r^{-1/d}\frac{\boldsymbol{t}}{\|\boldsymbol{t}\|})\big{)}\tilde{f}(r;\|\boldsymbol{t}\|^{d})r^{-\alpha}\mathrm{d}r$

where, for any $r>0$, according to (2.1)

$\tilde{f}(r;\lambda):=(\lambda r)^{1+\alpha}f(\lambda r)\ \to\
c_{f}\qquad(\lambda\to\infty).$ (2.8)

By boundedness of $\Xi^{0}$, ${\rm
Leb}_{d}\big{(}\Xi^{0}\cap(\Xi^{0}-r^{-1/d}\,\boldsymbol{z})\big{)}\leq{\rm
Leb}_{d}(\Xi^{0})$ is bounded and vanishes for $r>0$ small enough uniformly in
$\|\boldsymbol{z}\|=1$. Thus, (1) follows from (2) and (2.8) by the dominated
convergence theorem. Properties of $\ell(\boldsymbol{z})$ follow easily from
its definition and Assumption LRD.

(ii) Following (2), write

$\displaystyle\frac{c_{\lambda}(\phi)}{\lambda^{d(3-\alpha)}}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2d}}\phi(\boldsymbol{t}_{1})\phi(\boldsymbol{t}_{2})\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|^{d(1-\alpha)}\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}$
$\displaystyle\times\int_{0}^{\infty}{\rm
Leb}_{d}\Big{(}\Xi^{0}\cap\big{(}\Xi^{0}-r^{-1/d}\,\frac{\boldsymbol{t}_{1}-\boldsymbol{t}_{2}}{\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|}\big{)}\Big{)}\tilde{f}(r;\lambda\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|)r^{-\alpha}\mathrm{d}r$

and use (2.8) and the argument as in the proof of (i) to conclude (2.4).

The proof of (2.6) is similar but simpler. Indeed, by boundedness of $\Xi^{0}$
and (2.1) we have that $|r_{X}(\boldsymbol{t})|\leq
C(1\wedge\|\boldsymbol{t}\|^{d(1-\alpha)}),$ c.f. (2), so that the l.h.s. of
(2.6) does not exceed
$C\int_{\mathbb{R}^{2d}}|\phi(\boldsymbol{t}_{1}/\lambda)\phi(\boldsymbol{t}_{2}/\lambda)|(1\wedge\|\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\|^{2d(1-\alpha)})\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}$
whose evaluation by the r.h.s. in (2.6) for $\phi\in\Phi$ in (2.2) is
elementary. $\Box$

Introduce a Gaussian RF $B_{\alpha}(\phi)$ indexed by functions $\phi\in\Phi$
as stochastic integral

$\displaystyle B_{\alpha}(\phi)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}W_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t})\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t},\qquad\phi\in\Phi$ (2.9)

w.r.t. Gaussian white noise $W_{\alpha}$ with zero mean and variance
$\mathrm{E}W_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})^{2}=c_{f}r^{-1-\alpha}\mathrm{d}r\mathrm{d}\boldsymbol{u}$.
Observe that the variance

$\displaystyle\mathrm{E}B_{\alpha}(\phi)^{2}$ $\displaystyle=$ $\displaystyle
c_{f}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}r^{-1-\alpha}\mathrm{d}r\mathrm{d}\boldsymbol{u}\Big{(}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t})\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}\Big{)}^{2}$ (2.10) $\displaystyle=$
$\displaystyle
c_{f}\int_{\mathbb{R}^{2d}}\phi(\boldsymbol{t})\phi(\boldsymbol{t}^{\prime})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{t}^{\prime}\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0},\boldsymbol{t}^{\prime}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})r^{-1-\alpha}\mathrm{d}r\mathrm{d}\boldsymbol{u}$
$\displaystyle=$ $\displaystyle c(\phi)$

coincides with (2.5). Let be given a centered Poisson random measure
$\widetilde{N}_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})=N_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})-\mathrm{E}N_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})$
on $\mathbb{R}^{d}\times\mathbb{R}_{+}$ with variance
$\mathrm{E}\widetilde{N}_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})^{2}=\mathrm{E}N_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})=c_{f}r^{-1-\alpha}\mathrm{d}r\mathrm{d}\boldsymbol{u}$
the same as the variance of $W_{\alpha}$, and an $\alpha$-stable random
measure $L_{\alpha}(\mathrm{d}\boldsymbol{u})$ on $\mathbb{R}^{d}$ with the
characteristic function

$\displaystyle\mathrm{E}\mathrm{e}^{\mathrm{i}\theta L_{\alpha}(A)}$
$\displaystyle:=$ $\displaystyle\exp\big{\\{}{\rm
Leb}_{d}(A)\,c_{f}\int_{0}^{\infty}(\mathrm{e}^{\mathrm{i}\theta
r}-1-\mathrm{i}\theta r)r^{-1-\alpha}\mathrm{d}r\big{\\}}$ $\displaystyle=$
$\displaystyle\exp\big{\\{}-{\rm
Leb}_{d}(A)\,\sigma_{\alpha}|\theta|^{\alpha}\big{(}1-\mathrm{i}\,{\rm
sgn}(\theta)\,\tan(\pi\alpha/2)\big{)}\big{\\}},\quad\theta\in\mathbb{R},$

where $A\subset\mathbb{R}^{d}$ is any Borel set with ${\rm Leb}_{d}(A)<\infty$
and $\sigma_{\alpha}:=...$. Introduce RFs $L_{\alpha}$ and $J_{\alpha}$ as
stochastic integrals

$\displaystyle L_{\alpha}(\phi)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{d}}\phi(\boldsymbol{u})L_{\alpha}(\mathrm{d}\boldsymbol{u}),$
(2.12) $\displaystyle J_{\alpha}(\phi)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\widetilde{N}_{\alpha}(\mathrm{d}r,\mathrm{d}\boldsymbol{u})\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t})\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}$

w.r.t. to the above random measures which are well-defined for any
$\phi\in\Phi$. Clearly,
$\mathrm{E}J_{\alpha}(\phi)^{2}=\mathrm{E}B_{\alpha}(\phi)^{2}$. Denote
$\Psi(z):=\mathrm{e}^{\mathrm{i}z}-1-\mathrm{i}z,z\in\mathbb{R}$.

###### Theorem 1

Let
$X_{\lambda,M}(\phi)=\int_{\mathbb{R}^{d}}X_{M}(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t}$
with $X_{M}$ as in (1.5), $M=\lambda^{\gamma}\ (\gamma>0)$. Then for any
$\phi\in\Phi$ as $\lambda\to\infty$

$\displaystyle\lambda^{-H(\gamma)}(X_{\lambda,M}(\phi)-\mathrm{E}X_{\lambda,M}(\phi))$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}$
$\displaystyle\begin{cases}B_{\alpha}(\phi),&\gamma>d(\alpha-1),\
H(\gamma)=\frac{\gamma+(3-\alpha)d}{2},\\\
L_{\alpha}(\phi),&\gamma<d(\alpha-1),\ H(\gamma)=\frac{\gamma+d}{\alpha},\\\
J_{\alpha}(\phi),&\gamma=d(\alpha-1),\ H(\gamma)=d.\end{cases}$ (2.14)

Proof. Let
$j_{\lambda}(\theta):=\log\mathrm{E}\exp\\{\mathrm{i}\theta\lambda^{-H(\gamma)}(X_{\lambda,M}(\phi)-\mathrm{E}X_{\lambda,M}(\phi))\\},\theta\in\mathbb{R}$
denote the log-characteristic functional of the l.h.s. in (2.14). We need to
show that it converges to the corresponding functional $j(\theta)$ of the
r.h.s. as $\lambda\to\infty$.

Case $\gamma>d(\alpha-1)$. Set
$\tilde{j}_{\lambda}(\theta):=-(\theta^{2}/2){\rm
Var}(\lambda^{-H(\gamma)}X_{\lambda,M}(\phi)),$
$j(\theta):=\log\mathrm{E}\exp\\{\mathrm{i}\theta
B_{\alpha}(\phi)\\}=-(\theta^{2}/2)\mathrm{E}B_{\alpha}(\phi)^{2}$. Then
$\tilde{j}_{\lambda}(\theta)\to j(\theta)\ (\lambda\to\infty)$, see (2.4),
(2.5), (2.10). Next,

$\displaystyle j_{\lambda}(\theta)-\tilde{j}_{\lambda}(\theta)$
$\displaystyle=$
$\displaystyle\lambda^{\gamma}\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\mathrm{d}\boldsymbol{u}f(r)\mathrm{d}r\Big{\\{}\Psi\Big{(}\frac{\theta}{\lambda^{H(\gamma)}}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}/\lambda)\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}\Big{)}$ $\displaystyle\hskip
113.81102pt+(1/2)\Big{(}\frac{\theta}{\lambda^{H(\gamma)}}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}/\lambda)\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}\Big{)}^{2}\Big{\\}}$

Using $|\Psi(z)+(1/2)z^{2}|\leq|z|^{3}$ and (2.4) we get

$\displaystyle|j_{\lambda}(\theta)-\tilde{j}_{\lambda}(\theta)|$
$\displaystyle\leq$ $\displaystyle
C\lambda^{\gamma}\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\Big{|}\lambda^{-H(\gamma)}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}/\lambda)\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}\Big{|}^{3}\mathrm{d}\boldsymbol{u}f(r)\mathrm{d}r$
$\displaystyle\leq$ $\displaystyle
C\lambda^{-(\gamma/2)-(3/2)(3-\alpha)d}\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}/\lambda)|\mathrm{d}\boldsymbol{t}\,{\rm
Var}(X_{M}(\phi))$ $\displaystyle\leq$ $\displaystyle
C\lambda^{-(\gamma-d(\alpha-1))/2}\ =\ o(1)$

since
$|\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}/\lambda)\mathbb{I}(\boldsymbol{t}-\boldsymbol{u}\in
r^{1/d}\Xi^{0})\mathrm{d}\boldsymbol{t}|\leq\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}/\lambda)|\mathrm{d}\boldsymbol{t}<C\lambda^{d}$
uniformly in $\boldsymbol{u}\in\mathbb{R}^{d}$. Thus, $j_{\lambda}(\theta)\to
j(\theta)\,\forall\theta\in\mathbb{R}$.

Case $\gamma<d(\alpha-1)$. We have

$\displaystyle j_{\lambda}(\theta)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\Psi\Big{(}\theta\lambda^{\prime}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}+\boldsymbol{u})\mathbb{I}(\boldsymbol{t}\in(r/\lambda^{\prime})^{\frac{1}{d}}\,\Xi^{0})\mathrm{d}\boldsymbol{t}\Big{)}\tilde{f}(r;\lambda^{\frac{\gamma+d}{\alpha}})\frac{\mathrm{d}\boldsymbol{u}\mathrm{d}r}{r^{\alpha+1}},$
(2.15)

where $\lambda^{\prime}:=\lambda^{d-\frac{\gamma+d}{\alpha}}\to\infty$ and
$\tilde{f}$ is as in (2.8). Since $\phi$ is a.e. continuous and
$h_{\lambda}(\boldsymbol{u},r):=\lambda^{\prime}\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}+\boldsymbol{u})\mathbb{I}(\boldsymbol{t}\in(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}}\,\Xi^{0})\mathrm{d}\boldsymbol{t}\to
r\phi(\boldsymbol{u}){\rm Leb}_{d}(\Xi^{0})=:h(\boldsymbol{u},r)$ at each
continuity point $\boldsymbol{u}$ of $\phi$, we get that
$h_{\lambda}(\boldsymbol{u},r)\to h(\boldsymbol{u},r)$ for a.e.
$\boldsymbol{u}\in\mathbb{R}^{d}$ and each $r>0$. This fact together with
(2.15) and (2.8) suggest that

$\displaystyle j_{\lambda}(\theta)\ \to\ j(\theta)$ $\displaystyle=$
$\displaystyle
c_{f}\int_{\mathbb{R}^{d}\times\mathbb{R}_{+}}\big{(}\mathrm{e}^{\mathrm{i}\theta
r\phi(\boldsymbol{u}){\rm Leb}_{d}(\Xi^{0})}-1-\mathrm{i}\theta
r\phi(\boldsymbol{u}){\rm
Leb}_{d}(\Xi^{0})\big{)}\frac{\mathrm{d}\boldsymbol{u}\mathrm{d}r}{r^{\alpha+1}}$
$\displaystyle=$ $\displaystyle\log\mathrm{E}\exp\\{\mathrm{i}\theta
L_{\alpha}(\phi)\\},$

see (2), (2.12). The convergence in (2) can be justified using Pratt’s lemma
[32], as follows. It suffices to show that

$\displaystyle\int_{\mathbb{R}^{d}}\Psi(h_{\lambda}(\boldsymbol{u},r))\mathrm{d}\boldsymbol{u}$
$\displaystyle\to$
$\displaystyle\int_{\mathbb{R}^{d}}\Psi(h(\boldsymbol{u},r))\mathrm{d}\boldsymbol{u},\qquad\forall\,r>0,$
(2.17)
$\displaystyle\int_{\mathbb{R}^{d}}|\Psi(h_{\lambda}(\boldsymbol{u},r))|\mathrm{d}\boldsymbol{u}$
$\displaystyle\leq$ $\displaystyle C(r\wedge r^{2}),$ (2.18)
$\displaystyle\int_{0}^{\infty}(r\wedge
r^{2})\tilde{f}(r;\lambda^{\frac{\gamma+d}{\alpha}})\frac{\mathrm{d}r}{r^{\alpha+1}}$
$\displaystyle\to$ $\displaystyle c_{f}\int_{0}^{\infty}(r\wedge
r^{2})\frac{\mathrm{d}r}{r^{\alpha+1}}.$ (2.19)

Relations (2.17) and (2.19) are rather easy. To show (2.18), use
$|\Psi(z)|\leq 2(|z|\wedge|z|^{2})$ and boundedness of $\Xi^{0}$. Thus, the
l.h.s. of (2.18) does not exceed
$C\int_{\mathbb{R}^{d}}\min\Big{(}\lambda^{\prime}\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}+\boldsymbol{u})|\mathbb{I}(\|\boldsymbol{t}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t},\big{(}\lambda^{\prime}\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}+\boldsymbol{u})|\mathbb{I}(\|\boldsymbol{t}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t}\big{)}^{2}\Big{)}\mathrm{d}\boldsymbol{u}$
$\leq
C\min\Big{(}\lambda^{\prime}\int_{\mathbb{R}^{d}}(\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}+\boldsymbol{u})|\mathrm{d}\boldsymbol{u})\mathbb{I}(\|\boldsymbol{t}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t},(\lambda^{\prime})^{2}\int_{\mathbb{R}^{2d}}(\int_{\mathbb{R}^{d}}|\phi(\boldsymbol{t}_{1}+\boldsymbol{u})\phi(\boldsymbol{t}_{2}+\boldsymbol{u})|\mathrm{d}\boldsymbol{u})\mathbb{I}(\|\boldsymbol{t}_{1}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}},\|\boldsymbol{t}_{2}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t}_{1}\mathrm{d}\boldsymbol{t}_{2}\Big{)}$
$\leq
C\|\phi\|_{L^{1}}\big{(}\lambda^{\prime}\int_{\mathbb{R}^{d}}\mathbb{I}(\|\boldsymbol{t}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t}\big{)}\wedge\big{(}\lambda^{\prime}\int_{\mathbb{R}^{d}}\mathbb{I}(\|\boldsymbol{t}\|\leq
C(\frac{r}{\lambda^{\prime}})^{\frac{1}{d}})\mathrm{d}\boldsymbol{t}\big{)}^{2}\leq
C(r\wedge r^{2})$. This proves (2.18) and (2).

Case $\gamma=d(\alpha-1)$. The expression in (2.15) with $\lambda^{\prime}=1$
for $j_{\lambda}(\theta)$ is valid. The proof of (2.14) in this case is
similar to that when $\gamma<d(\alpha-1)$ and we omit the details. $\Box$

###### Remark 1

Theorem 1 applies also to $\gamma=0$ or integrals in (1.1) of the RG model in
(1.4) with Poisson intensity $\mathrm{d}\boldsymbol{u}F(\mathrm{d}r)$. Indeed,
the argument in the case $\gamma<d(\alpha-1)$ applies without change when
$\gamma=0$, yielding the stable limit

$\displaystyle\lambda^{-d/\alpha}(X_{\lambda}(\phi)-\mathrm{E}X_{\lambda}(\phi))$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}$ $\displaystyle
L_{\alpha}(\phi),\qquad\forall\,\phi\in\Phi.$ (2.21)

###### Remark 2

Condition $M=\lambda^{\gamma}$ in Theorem 1 can be weakened [15].
Particularly, it can be replaced by $M=y\lambda^{\gamma}$ for some $y>0$, in
which case the control measure of the limit RFs in (2.14) contain the extra
multiplicative factor $y$. For
$d=1,\phi_{s}(t):=\mathbb{I}(t\in]0,s]),t\in\mathbb{R},(s,y)\in\mathbb{R}^{2}_{+}$
we have that $X_{\lambda,M}(\phi_{s})=\int_{0}^{\lambda
s}X_{M}(t)\mathrm{d}t$, where
$X_{M}(t)=\sum_{j\in\mathbb{Z}}\mathbb{I}(u_{j}(M)<t\leq u_{j}(M)+R_{j})$ is
the number of customers at time $t$ in the M/G/$\infty$ queue with service
time distribution $R$ and a Poisson arrival process
$\\{u_{j}(M);j\in\mathbb{Z}\\}$ with intensity
$M\mathrm{d}u=y\lambda^{\gamma}\mathrm{d}u$. Then

$\displaystyle\lambda^{-H(\gamma)}\int_{0}^{\lambda
s}(X_{M}(t)-\mathrm{E}X_{M}(t))\mathrm{d}t$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{\longrightarrow}}\\\ \end{array}$
$\displaystyle\begin{cases}B_{\alpha}(s,y),&\gamma>\alpha-1,\
H(\gamma)=\frac{\gamma+3-\alpha}{2},\\\ L_{\alpha}(s,y),&\gamma<\alpha-1,\
H(\gamma)=\frac{\gamma+1}{\alpha},\\\ J_{\alpha}(s,y),&\gamma=\alpha-1,\
H(\gamma)=1,\end{cases}$ (2.23)

where the limits are RFs indexed by $(s,y)\in\mathbb{R}^{2}_{+}$ written as
stochastic integrals w.r.t. Gaussian, $\alpha$-stable and Poisson random
measures on $\mathbb{R}\times\mathbb{R}^{2}_{+}$ analogously to (2.9), (2.12).
See [19] for details. Particularly,
$\\{B_{\alpha}(s,v);(s,v)\in\mathbb{R}^{2}_{+}\\}$ is a fractional Brownian
sheet with Hurst parameters $(H_{1},H_{2})=(\frac{3-\alpha}{2},\frac{1}{2})$,
$\\{L_{\alpha}(s,v);(s,v)\in\mathbb{R}^{2}_{+}\\}$ is an $\alpha$-stable Lévy
sheet, and $\\{J_{\alpha}(s,v);(s,v)\in\mathbb{R}^{2}_{+}\\}$ is the Telecom
RF defined in [19] as a RF extension of the corresponding Telecom process in
[16]. As noted above, Theorem 1 is essentially due to [15] while (2.23) is a
version of the results in [22, 16] and other previous work.

##  3 Charlier polynomials and Mehler’s formula

The derivations in this sec. can be compared to the discussion of Hermite
polynomials and expansions in the case of Gaussian distribution [13,
pp.22-26]. See [9] for classical Mehler’s formula for Hermite polynomials.

Recall from [39, 24] the definition of Charlier polynomials $P_{k}(x;\mu)$ of
discrete variable $x\in\mathbb{N}$ through the generating function:

${\cal
P}(u;x,\mu):=\sum_{k=0}^{\infty}\frac{u^{k}}{k!}P_{k}(x;\mu)=(1+u)^{x}\mathrm{e}^{-u\mu},$
(3.1)

where the series is convergent for any $x\in\mathbb{N},\mu>0$ and any
(complex) $u\in\mathbb{C}$. We have
$P_{0}(x;\mu)=1,P_{1}(x;\mu)=x-\mu,P_{2}(x;\mu)=x^{2}-(2\mu+1)x+\mu^{2}$ and

$P_{k}(x;\mu)=(-1)^{k}\mu^{k}p(x;\mu)^{-1}D_{-}^{k}p(x;\mu),\qquad
k\in\mathbb{N}$ (3.2)

where $D_{-}^{k}:=D_{-}D_{-}^{k-1}$ is the backward difference operator,
$D_{-}G(x):=G(x)-G(x-1)\mathbb{I}(x\geq 1),D^{0}_{-}G(x)=G(x)$ and

$p(x;\mu)=\mathrm{e}^{-\mu}\frac{\mu^{x}}{x!},\quad x\in\mathbb{N}$ (3.3)

is the distribution of Poisson r.v. $N$ with mean $\mu$. Relation (3.2)
follows from (3.1) using the identity $(1+u)\partial{\cal P}(u;x,\mu)/\partial
u=(x-(1+u)\mu){\cal P}(u;x,\mu)$ [39]. We have

$\displaystyle\mathrm{E}P_{k}(N;\mu)=0,\qquad\mathrm{E}P_{k}(N;\mu)^{2}=k!\mu^{k},\quad
k=1,2,\cdots,$ (3.4)
$\displaystyle\mathrm{E}P_{k}(N;\mu)P_{\ell}(N;\mu)=0,\quad
k\neq\ell=0,1,\cdots.$

Facts (3.4) follow from multiplying the series in (3.1) at the points $u$ and
$v$ and taking the expectation of the product:

$\displaystyle\sum_{k,\ell=0}^{\infty}\frac{u^{k}v^{\ell}}{k!\ell!}\mathrm{E}P_{k}(N;\mu)P_{\ell}(N;\mu)$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-(u+v)\mu}\mathrm{E}[((1+u)(1+v))^{N}]$
$\displaystyle=$ $\displaystyle\mathrm{e}^{\mu uv}\ =\
\sum_{k=0}^{\infty}\frac{(\mu uv)^{k}}{k!}$

and equating the coefficients of $u^{k}v^{\ell},k,\ell\in\mathbb{N}$ of the
power series on both sides.

Any $G=G(x),x\in\mathbb{N}$ with $\mathrm{E}G^{2}(N)<\infty$ can be uniquely
expanded in Charlier polynomials

$G(x)=\sum_{k=0}^{\infty}\frac{c_{G}(k;\mu)}{k!}P_{k}(x;\mu),\qquad
x\in\mathbb{N}$ (3.5)

where

$c_{G}(k;\mu):=\mu^{-k}\mathrm{E}G(N)P_{k}(N;\mu),\quad k\in\mathbb{N}$ (3.6)

are Charlier coefficients of $G$ in (4.13). (3.2) and summation by parts
yields another expression for these coefficients

$c_{G}(k;\mu)=\mathrm{E}D_{+}^{k}G(N),\quad k\in\mathbb{N},$ (3.7)

where $D^{k}_{+}:=D_{+}D_{+}^{k-1}$ is the forward difference operator,
$D_{+}G(x):=G(x+1)-G(x),D^{2}_{+}G(x)=D_{+}G(x+1)-D_{+}G(x)=G(x+2)-2G(x+1)+G(x)$
etc. (3.6) and (3.4) yield the bound

$|c_{G}(k;\mu)|\leq\mu^{-k}\sqrt{\mathrm{E}[G^{2}(N)]\mathrm{E}[P^{2}_{k}(N;\mu)]}=C(k!/\mu^{k})^{1/2},\quad
C=\sqrt{\mathrm{E}[G(N)^{2}]}.$ (3.8)

###### Lemma 1

Let $M_{i},i=1,2,3,$ be independent Poisson distributed r.v.s with respective
means
$\mathrm{E}M_{1}=\mu_{1}-\mu_{3},\,\mathrm{E}M_{2}=\mu_{2}-\mu_{3},\,\mathrm{E}M_{3}=\mu_{3},$
$0\leq\mu_{3}<\mu_{1}\wedge\mu_{2}$, and

$N_{i}:=M_{i}+M_{3},\quad i=1,2.$

Let

$p(x,y;\mu_{1},\mu_{2},\mu_{3}):=\mathrm{P}(N_{1}=x,N_{2}=y),\qquad(x,y)\in\mathbb{N}^{2}$
(3.9)

denote the joint distribution of $(N_{1},N_{2})$.

(i) (Orthogonality property): For any $k,\ell\in\mathbb{N}$

$\displaystyle\mathrm{E}P_{k}(N_{1};\mu_{1})P_{\ell}(N_{2};\mu_{2})=\begin{cases}0,&k\neq\ell,\\\
\mu^{k}_{3}k!,&k=\ell,\end{cases}$ (3.10)

with the convention $0^{0}:=1$.

(ii) Let $G_{i}=G_{i}(x),x\in\mathbb{N},i=1,2$ be given functions such that

$\mathrm{E}G^{2}_{i}(N_{i})<\infty,\qquad i=1,2.$ (3.11)

Then

$\displaystyle\mathrm{E}G_{1}(N_{1})G_{2}(N_{2})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{c_{G_{1}}(k;\mu_{1})c_{G_{2}}(k;\mu_{2})}{k!}\mu_{3}^{k}.$
(3.12)

(iii) (Mehler’s formula):

$\displaystyle p(x,y;\mu_{1},\mu_{2},\mu_{3})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\mu^{k}_{3}}{k!}D^{k}_{-}p(x;\mu_{1})D^{k}_{-}p(y;\mu_{2})$
$\displaystyle=$ $\displaystyle
p(x;\mu_{1})p(y;\mu_{2})\sum_{k=0}^{\infty}\frac{\rho_{12}^{k}}{k!}P_{k}(x;\mu_{1})P_{k}(y;\mu_{2}),$

where $\rho_{12}:=\mu_{3}/\sqrt{\mu_{1}\mu_{2}}={\rm Corr}(N_{1},N_{2})$ is
the correlation coefficient.

Proof. (i) The proof of (3.10) using the generating function in (3.2) is
similar as in the univariate case of (3.4). Consider the expectation

$\displaystyle\mathrm{E}{\cal P}(u;N_{1},\mu_{1}){\cal P}(v;N_{2},\mu_{2})$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-u\mu_{1}-v\mu_{2}}\mathrm{E}[(1+u)^{N_{1}}(1+v)^{N_{2}}]$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-u\mu_{1}-v\mu_{2}}\mathrm{E}[(1+u)^{M_{1}}]\mathrm{E}[(1+v)^{M_{2}}]\mathrm{E}[((1+u)(1+v))^{M_{3}}]$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-u\mu_{1}-v\mu_{2}}\mathrm{e}^{(\mu_{1}-\mu_{3})u}\mathrm{e}^{(\mu_{2}-\mu_{3})v}\mathrm{e}^{((1+u)(1+v)-1)\mu_{3}}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{uv\mu_{3}}=\sum_{k=0}^{\infty}\frac{(uv\mu_{3})^{k}}{k!}.$

On the other hand,

$\displaystyle\mathrm{E}{\cal P}(u;N_{1},\mu_{1}){\cal P}(v;N_{2},\mu_{2})$
$\displaystyle=$
$\displaystyle\sum_{k,\ell=0}^{\infty}\frac{u^{k}v^{\ell}}{k!\ell!}\mathrm{E}[P_{k}(N_{1};\mu_{1})P_{\ell}(N_{2};\mu_{2})]$

and (3.10) follows by equating the coefficients of
$u^{k}v^{\ell},k,\ell\in\mathbb{N}$ of the power series on both sides.

(ii) Immediate from (4.13) and (3.10).

(iii) Apply (3.12) to $G_{1}(x):=\mathbb{I}(x=n),G_{2}(x):=\mathbb{I}(x=m),$
for given $n,m\in\mathbb{N}$. By (3.7), (3.2),
$c_{G_{1}}(k;\mu_{1})=\mathrm{E}[D^{k}_{+}\mathbb{I}(N_{1}=n)]=D^{k}_{-}p(n;\mu_{1})=(-1)^{k}\mu_{1}^{-k}P_{k}(n;\mu_{1})p(n;\mu_{1}),$
$c_{G_{2}}(k;\mu_{2})=\mathrm{E}[D^{k}_{+}\mathbb{I}(N_{2}=m)]=D^{k}_{-}p(m;\mu_{2})=(-1)^{k}\mu_{2}^{-k}P_{k}(m;\mu_{2})p(m;\mu_{2}).$
This proves (1) and the lemma, too. $\Box$

###### Remark 3

Let $N=\\{N_{t};t=0,1,\cdots\\}$ be a stationary Markov process on
$\mathbb{N}$ with marginal Poisson distribution $\mathrm{P}(N_{t}=x)=p(x;\mu)$
and transition probabilities

$p(y|x;\mu):=\frac{p(x,y;\mu)}{p(x;\mu)},\qquad x,y\in\mathbb{N},$ (3.15)

where $p(x,y;\mu):=p(x,y;\mu,\mu,\mu_{3})$ is the joint distribution in (3.9)
with $\mu_{1}=\mu_{2}=:\mu>\mu_{3}$. This process is well-known in the
literature as the Poisson AR(1) or INAR(1) and is related to M/M/$\infty$
queueing system, see e.g. [21]. Substitution of (1) into (3.15) yields an
expansion of (3.15) into a series of Charlier polynomials. Since the
transition probability of the Poisson INAR(1) process is usually written via a
different expansion, the coincidence with (3.15) most easily can be verified
through the bivariate generating function of $(N_{1},N_{2})$ in (3), see [21,
(9)]. We remark that the Poisson INAR(1) process is a particular case of
stationary Markov evolutions of non-interacting particle systems with Poisson
marginal distribution discussed in [39], closely related to chaos expansions
in multiple Poisson stochastic integrals.

Given a $G(x),x\in\mathbb{N},\mathrm{E}G(N)^{2}<\infty$ with Charlier
expansion in (4.13) we define the Charlier rank $k^{*}(G;\mu)$ of $G$ as the
minimal $k\geq 1$ such that $c_{G}(k;\mu)\neq 0$, viz.,

$k^{*}(G;\mu):=\min\\{k\geq 1:c_{G}(k;\mu)\neq 0\\}.$

Lemma 1 (3.12) and the bound in (3.8) imply the following

###### Corollary 1

Let $G_{i},N_{i},i=1,2,$ be as in Lemma 1, $\rho_{12}={\rm
Corr}(N_{1},N_{2})$, $k^{*}:=k_{C}(G_{1};\mu_{1})\vee k_{C}(G_{2};\mu_{2})$.
Then

$\displaystyle{\rm Cov}(G_{1}(N_{1}),G_{2}(N_{2}))$ $\displaystyle=$
$\displaystyle\sum_{k=k^{*}}^{\infty}\frac{c_{G_{1}}(k;\mu_{1})c_{G_{2}}(k;\mu_{2})}{k!}\mu_{3}^{k}$
$\displaystyle=$
$\displaystyle\frac{c_{G_{1}}(k^{*};\mu_{1})c_{G_{2}}(k^{*};\mu_{2})}{k^{*}!}\mu_{3}^{k^{*}}+R(k^{*}),$

where

$|R(k^{*})|\ \leq\
\frac{(\mu_{3}/\sqrt{\mu_{1}\mu_{2}})^{k^{*}+1}}{1-(\mu_{3}/\sqrt{\mu_{1}\mu_{2}})}\prod_{i=1}^{2}\mathrm{E}^{1/2}[G(N_{i})^{2}].$

Moreover,

$\displaystyle|{\rm Cov}(G_{1}(N_{1}),G_{2}(N_{2}))|$ $\displaystyle\leq$
$\displaystyle\sum_{k=k^{*}}^{\infty}|\rho_{12}|^{k}\frac{|c_{G_{1}}(k;\mu_{1})c_{G_{2}}(k;\mu_{2})|(\mu_{1}\mu_{2})^{k/2}}{k!}$
$\displaystyle\leq$
$\displaystyle|\rho_{12}|^{k^{*}}\prod_{i=1}^{2}\big{(}\sum_{k=k^{*}}^{\infty}\frac{c^{2}_{G_{i}}(k;\mu_{i})\mu_{i}^{k}}{k!}\big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle|\rho_{12}|^{k^{*}}\sqrt{{\rm
Var}(G_{1}(N_{1})){\rm Var}(G_{2}(N_{2}))}.$

Particularly, $\sup\\{|{\rm Cov}(G_{1}(N_{1}),G_{2}(N_{2}))|:{\rm
Var}(G_{i}(N_{i}))=1,i=1,2\\}=|\rho_{12}|$ and the last supremum is attained
by linear functions $G_{i}(x)=x/\sqrt{\mu_{i}},i=1,2$.

##  4 Scaling limits of nonlinear functions of RG model

We study the limit distribution of $Y_{M}(\phi),Y_{\lambda,M}(\phi)$ defined
in (1.6)-(1.7) as $\lambda\to\infty$ and $M=\lambda^{\gamma}\to\infty$, for
some $\gamma>0$, where $G=G(x),x\in\mathbb{R}$ is a general function
satisfying some conditions. We will show that the limit of
$Y_{\lambda,M}(\phi)$ is the same as that of the linear integral
$X_{\lambda,M}(\phi)$ in Theorem 1 up to the multiplicative constant equal to
the first coefficient in the Hermite expansion of $G$. Since for each
$\boldsymbol{t}\in\mathbb{R}^{d}$, the quantity inside $G$ in
$Y_{M}(\boldsymbol{t})$ of (1.6), viz.,
$(X_{M}(\boldsymbol{t})-\mathrm{E}X_{M}(\boldsymbol{t}))/M^{1/2}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}Z_{\mu}\sim N(0,\mu)$ when $M\to\infty$,
we consider the corresponding Hermite expansion

$G(x)=\sum_{k=0}^{\infty}\frac{h_{G,\mu}(k)}{k!}H_{k}(x;\mu)$ (4.1)

in Hermite polynomials $H_{k}(x;\mu),k\in\mathbb{N},x\in\mathbb{R}$ with
generating function
$\sum_{k=0}^{\infty}(u^{k}/k!)H_{k}(x;\mu)=\mathrm{e}^{ux-\mu u^{2}/2}$ and
Hermite coefficients

$\displaystyle
h_{G,\mu}(k):=\mu^{-k}\mathrm{E}[G(Z_{\mu})H_{k}(Z_{\mu};\mu)],\qquad
k\in\mathbb{N}.$ (4.2)

###### Theorem 2

Let $Y_{\lambda,M}(\phi)$ be as in (1.6), where $X_{M}(\boldsymbol{t})$
satisfies the conditions of Theorem 1 and $G=G(x),x\in\mathbb{R}$ is a Borel
function such that

$\lim_{M\to\infty}\mathrm{E}G\Big{(}\frac{X_{M}(\boldsymbol{0})-\mathrm{E}X_{M}(\boldsymbol{0})}{M^{1/2}}\Big{)}^{2}=\mathrm{E}G(Z_{\mu})^{2}<\infty.$
(4.3)

Let $M=\lambda^{\gamma}$ for some $\gamma>0$. Then for any $\phi\in\Phi$ as
$\lambda\to\infty$

$\displaystyle\lambda^{(\gamma/2)-H(\gamma)}(Y_{\lambda,M}(\phi)-\mathrm{E}Y_{\lambda,M}(\phi))$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}$ $\displaystyle
h_{G,\mu}(1)\begin{cases}B_{\alpha}(\phi),&\gamma>d(\alpha-1),\\\
L_{\alpha}(\phi),&\gamma<d(\alpha-1),\\\
J_{\alpha}(\phi),&\gamma=d(\alpha-1),\end{cases}$ (4.5)

where $H(\gamma),B_{\alpha}(\phi),L_{\alpha}(\phi),J_{\alpha}(\phi)$ are the
same as in (2.14), and

$h_{G,\mu}(1)=\mu^{-1}\mathrm{E}G(Z_{\mu})Z_{\mu}$ (4.6)

is the first coefficient in the Hermite expansion (4.1) of $G$.

Proof. We have $Y_{M}(\boldsymbol{t})=G_{M}(X_{M}(\boldsymbol{t}))$ where
$G_{M}(x):=G((x-\mu M)/M^{1/2}),x\in\mathbb{N}$ and
$X_{M}(\boldsymbol{t})=:N_{M}$ has a Poisson distribution with mean $\mu M$.
Following (4.13) consider the expansion

$G_{M}(x)=\sum_{k=0}^{\infty}\frac{c_{G,M}(k)}{k!}P_{k}(x;\mu M),\qquad
x\in\mathbb{N}$ (4.7)

where

$c_{G,M}(k):=(\mu M)^{-k}\mathrm{E}[G_{M}(N_{M})P_{k}(N_{M};\mu M)],\quad
k\in\mathbb{N}$ (4.8)

Particularly,

$\displaystyle M^{1/2}c_{G,M}(1)$ $\displaystyle=$
$\displaystyle\frac{M^{1/2}}{\mu
M}\mathrm{E}\Big{[}G\big{(}\frac{N_{M}-M\mu}{M^{1/2}}\big{)}(N_{M}-M\mu)\Big{]}\
\to\ h_{G,\mu}(1)$ (4.9)

as $M\to\infty$, the limit as in (4.6). The convergence in (4.9) under
condition (4.3) can be verified with the help of Pratt’s lemma [32], as
follows. Let
$g(x):=G(x)x,x\in\mathbb{R},\,\xi_{M}:=\frac{N_{M}-M\mu}{M^{1/2}}$, then
$M^{1/2}c_{G,M}(1)=\mu^{-1}\mathrm{E}g(\xi_{M})\xi_{M}$ and
$\xi_{M}\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}Z_{\mu}\,(M\to\infty)$. Since $G$ is a.e.
continuous and $Z_{\mu}$ has a continuous distribution, this implies
$G(\xi_{M})\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}G(Z_{\mu}),\,g(\xi_{M})$
$\begin{array}[t]{c}\stackrel{{\scriptstyle\rm d}}{{\longrightarrow}}\\\
\end{array}g(Z_{\mu})$. By the Skorohod representation theorem,
$\xi_{M}\to_{p}Z_{\mu},G(\xi_{M})\to_{p}G(Z_{\mu}),g(\xi_{M})\to_{p}g(Z_{\mu})$
in probability. Moreover,
$|g(\xi_{M})|\leq(1/2)(G(\xi_{M})^{2}+\xi_{M}^{2})=:\bar{g}_{M}$ where
$\bar{g}_{M}\to_{p}(1/2)(G(Z_{\mu})^{2}$ $+Z^{2}_{\mu})=:\bar{g}$ and
$\mathrm{E}\bar{g}_{M}\to\mathrm{E}\bar{g}$ according to (4.3). This and [32]
prove (4.9).

In view of (4.7), we have the representation

$\displaystyle Y_{\lambda,M}(\phi)-\mathrm{E}Y_{\lambda,M}(\phi)$
$\displaystyle=$ $\displaystyle
c_{G,M}(1)(X_{\lambda,M}(\phi)-\mathrm{E}X_{\lambda,M}(\phi))+Y^{*}_{\lambda,M}(\phi),\qquad\text{where}$
(4.10) $\displaystyle Y^{*}_{\lambda,M}(\phi)$ $\displaystyle:=$
$\displaystyle\int_{\mathbb{R}^{d}}Y^{*}_{M}(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t},\qquad
Y^{*}_{M}(\boldsymbol{t}):=\sum_{k=2}^{\infty}\frac{c_{G,M}(k)}{k!}P_{k}\big{(}X_{M}(\boldsymbol{t});\mu
M\big{)}.$

The convergence in (2.14) follows from (4.5) and (4.9), once we show that
$Y^{*}_{\lambda,M}(\phi)$ in (4.10) is negligible, or

$\displaystyle\mathrm{E}Y^{*}_{\lambda,M}(\phi)^{2}$ $\displaystyle=$
$\displaystyle o(\lambda^{2H(\gamma)-\gamma}),\qquad\lambda\to\infty$ (4.11)

for $M,H(\gamma)$ as in Theorem 2 and any fixed $\gamma>0$. Applying Corollary
1 with $k^{*}=2,\mu_{3}={\rm
Cov}(X_{M}(\boldsymbol{t}),X_{M}(\boldsymbol{0}))=Mr_{X}(\boldsymbol{t})$ and
the bound $c^{2}_{G,M}(k)\leq(\mu
M)^{-k}k!\mathrm{E}G_{M}(X_{M}(\boldsymbol{0}))^{2}$, see (3.8), we get

$\displaystyle|\mathrm{E}Y^{*}_{M}(\boldsymbol{t})Y^{*}_{M}(\boldsymbol{0})|$
$\displaystyle\leq$
$\displaystyle\sum_{k=2}^{\infty}\frac{c^{2}_{G,M}}{(k!)^{2}}\big{|}\mathrm{E}[P_{k}(X_{M}(\boldsymbol{t});\mu
M)P_{k}(X_{M}(\boldsymbol{0});\mu M)]\big{|}$ (4.12) $\displaystyle\leq$
$\displaystyle\mathrm{E}G_{M}(X_{M}(\boldsymbol{0}))^{2}\sum_{k=2}^{\infty}\frac{(r_{X}(\boldsymbol{t})M)^{k}}{(\mu
M)^{k}}$ $\displaystyle=$
$\displaystyle\mathrm{E}G_{M}(X_{M}(\boldsymbol{0}))^{2}\sum_{k=2}^{\infty}\big{(}\frac{r_{X}(\boldsymbol{t})}{r_{X}(\boldsymbol{0})}\big{)}^{k}\
\leq\ C(1\wedge r^{2}_{X}(\boldsymbol{t})).$

Applying Proposition 1 (2.6), relation (4.11) follows since
$\max\\{d,2d(2-\alpha)\\}<\min\\{(3-\alpha)d,\frac{2(\gamma+d)}{\alpha}-\gamma\\}$
holds for any $\gamma>0,\alpha\in(1,2)$. $\Box$

###### Remark 4

Condition (4.3) on $G$ involving convergence of the second moments only is
rather weak. Using the notation in (1.6), it writes as
$\lim_{M\to\infty}\mathrm{E}Y_{M}(0)^{2}=\mathrm{E}G(Z_{\mu})^{2}<\infty.$
(4.3) can be replaced by a boundedness condition:

$\displaystyle|G(x)|\leq C_{1}\mathrm{e}^{C_{2}|x|},\quad
x\in\mathbb{R}\quad(\exists\,C_{1},C_{2}>0).$ (4.13)

Indeed, verification of (4.3) for $G(x)=C_{1}\mathrm{e}^{C_{2}|x|}$ is easy,
implying (4.3) for $G$ in (4.13) by Pratt’s lemma.

Obviously, Theorem 2 does not hold when $\gamma=0$ or $M=1$ is fixed. This
case is treated in the following proposition.

###### Proposition 2

Let
$Y(\boldsymbol{t}):=G\big{(}X(\boldsymbol{t})\big{)},\boldsymbol{t}\in\mathbb{R}^{d},$
where $X(\boldsymbol{t})$ is as in (1.4) and $G(x),x\in\mathbb{N}$ satisfies
$\mathrm{E}G(X(\boldsymbol{0})^{2}<\infty$,
$Y_{\lambda}(\phi):=\int_{\mathbb{R}^{d}}Y(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t},\phi\in\Phi.$
Then

$\displaystyle\lambda^{-d/\alpha}(Y_{\lambda}(\phi)-\mathrm{E}Y_{\lambda}(\phi))$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\ \end{array}$ $\displaystyle
c_{G}(1;\mu)L_{\alpha}(\phi),$ (4.15)

where
$c_{G}(1;\mu)=\mathrm{E}G(X(\boldsymbol{0}))(X(\boldsymbol{0})-\mathrm{E}X(\boldsymbol{0}))$
and $L_{\alpha}(\phi)$ is the same $\alpha$-stable RF as in (2.21).

Proof. Similarly as in the proof of Theorem 2, write
$Y_{\lambda}(\phi)-\mathrm{E}[Y_{\lambda}(\phi)]=c_{G}(1;\mu)(X_{\lambda}(\phi)-\mathrm{E}[X_{\lambda}(\phi)])+$
$Y^{*}_{\lambda}(\phi),$ where
$Y^{*}_{\lambda}(\phi):=\int_{\mathbb{R}^{d}}Y^{*}(\boldsymbol{t})\phi(\boldsymbol{t}/\lambda)\mathrm{d}\boldsymbol{t}$
and
$Y^{*}(\boldsymbol{t}):=\sum_{k=2}^{\infty}\frac{c_{G}(k;\mu)}{k!}P_{k}\big{(}X(\boldsymbol{t});\mu\big{)},$
$\mu=\mathrm{E}[X(\boldsymbol{0})]$ satisfies
$|\mathrm{E}Y^{*}(\boldsymbol{t})Y^{*}(\boldsymbol{0})|\leq C(1\wedge
r^{2}_{X}(\boldsymbol{t}))$ as in (4.12). Then, (4.15) follows in view of
(2.21) and (2.6). $\Box$

###### Remark 5

Theorem 2 and Proposition 2 yield trivial limits if the respective
coefficients $h_{G,\mu}(1),$ $c_{G}(1)$ vanish. The question of the limit
distribution of $Y_{\lambda,M}(\phi),Y_{\lambda}(\phi)$ in such case is open.

###### Example 1

(Scaling limit of the Boolean model. ) The Boolean model
$\hat{X}(\boldsymbol{t})=X(\boldsymbol{t})\wedge 1$ corresponds to
$Y(\boldsymbol{t})=G(X(\boldsymbol{t}))$ with $G(x)=x\wedge 1,x\in\mathbb{N}$.
We have
$c_{G}(0;\mu)=1-\mathrm{e}^{-\mu},c_{G}(k;\mu)=(-1)^{k+1}\mathrm{e}^{-\mu}\,(k\geq
1)$ and the convergence in (4.15) holds with $c_{G}(1;\mu)=\mathrm{e}^{-\mu}$.
Let $\phi(\boldsymbol{x})=\mathbb{I}(x\in A),$ where $A\subset\mathbb{R}^{d}$
is a Borel set and $\hat{X}_{\lambda}(A)$ be as in (1.9). From Proposition 2
and (2.21) it follows

###### Corollary 2

Let $A\subset\mathbb{R}^{d}$ be a bounded Borel set and $\hat{X}_{\lambda}(A)$
as in (1.9). Then

$\lambda^{-d/\alpha}(\hat{X}_{\lambda}(A)-\mathrm{E}\hat{X}_{\lambda}(A))\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
d}}{{\longrightarrow}}\\\
\end{array}\mathrm{e}^{-\mu}L_{\alpha}(A),\quad\lambda\to\infty$ (4.16)

where $L_{\alpha}(A)$ is $\alpha$-stable r.v. with characteristic function
$\mathrm{E}\mathrm{e}^{\mathrm{i}\theta
L_{\alpha}(A)}=\exp\\{-\sigma_{\alpha}|\theta|^{\alpha}{\rm
Leb}_{d}(A)(1-\mathrm{i}\,{\rm
sgn}(\theta)\,\tan(\pi\alpha/2))\\},\theta\in\mathbb{R}.$

###### Example 2

(Scaling limits of the Exponential RG model. ) We define the Exponential RG
model as ${\cal
E}(\boldsymbol{t}):=\mathrm{e}^{aX(\boldsymbol{t})},\boldsymbol{t}\in\mathbb{R}^{d}$
where $X(\boldsymbol{t})$ is the RG model in (1.4) and $a\in\mathbb{R}$ a real
parameter. We also consider the exponential function of the aggregated RG
model

$\displaystyle{\cal E}_{M}(\boldsymbol{t})$ $\displaystyle:=$
$\displaystyle\mathrm{e}^{a(X_{M}(\boldsymbol{t})-\mathrm{E}X_{M}(\boldsymbol{t}))/M^{1/2}},\qquad{\cal
E}_{\lambda,M}(\phi):=\int_{\mathbb{R}^{d}}\phi(\boldsymbol{t}/\lambda){\cal
E}_{M}(\boldsymbol{t})\mathrm{d}\boldsymbol{t},$ (4.17)

where $X_{M},\phi$ are as in (1.5). As noted in the Introduction, the interest
in the scaling limits of (4.17) is motivated by application to large-time
asymptotics of statistical solution of Burgers’ equation discussed in the last
section. Obviously, (4.17) is a particular case of (1.6) corresponding to
$G(x)=\mathrm{e}^{ax}$. Note
$D^{k}_{+}G(x)=(\mathrm{e}^{a}-1)^{k}\mathrm{e}^{ax}$ and
$c_{G}(k)=(\mathrm{e}^{a}-1)^{k}\mathrm{e}^{(\mathrm{e}^{a}-1)\mu},k\in\mathbb{N}$.
We also find that

$\displaystyle M^{1/2}c_{G,M}(1)$ $\displaystyle=$
$\displaystyle\exp\\{(\mathrm{e}^{a/M^{1/2}}-1-(a/M^{1/2}))\mu
M\\}M^{1/2}(\mathrm{e}^{a/M^{1/2}}-1)$ (4.18) $\displaystyle\to$
$\displaystyle a\mathrm{e}^{a^{2}\mu/2}\ =\
\mu^{-1}\mathrm{E}[\mathrm{e}^{aZ_{\mu}}Z_{\mu}]=h_{G,\mu}(1)$

as $M\to\infty$, see (4.6). It is easy to see $G(x)=\mathrm{e}^{ax}$ that
satisfies the conditions of Theorem (2) and the convergences in (4.5) hold for
(4.17) with $h_{G,\mu}(1)$ in (4.18). The relevant bound in (4.12) for the
above $G(x)$ can be directly obtained from the equality

$\displaystyle{\rm Cov}\big{(}{\cal E}^{*}_{M}(\boldsymbol{0}),{\cal
E}^{*}_{M}(\boldsymbol{t})\big{)}$ $\displaystyle=$
$\displaystyle(\mathrm{E}{\cal
E}_{M}(\boldsymbol{0}))^{2}\big{\\{}\mathrm{e}^{(\mathrm{e}^{a/M^{1/2}}-1)^{2}Mr_{X}(\boldsymbol{t})}-1-(\mathrm{e}^{a/M^{1/2}}-1)^{2}Mr_{X}(\boldsymbol{t})\big{\\}},$

where ${\cal E}^{*}_{M}(\boldsymbol{t})=({\cal
E}_{M}(\boldsymbol{t})-\mathrm{E}{\cal
E}_{M}(\boldsymbol{t}))-c_{G,M}(1)(X_{M}(\boldsymbol{t})-\mathrm{E}X_{M}(\boldsymbol{t}))$
as in (4.12).

###### Remark 6

[3] discusses small-scale scaling limits of RG model in (1.5) as $\lambda\to
0$ and $M\to 0$ together with $\lambda$, under a similar condition (c.f.
(2.1)) on the behavior of $f(r)$ as $r\to 0$ with $\alpha\in(0,1)$. Extending
these results to nonlinear functions in (1.6) is open. Anisotropic small-scale
limits (without aggregation) for Lévy driven RFs on $\mathbb{R}^{2}$ were
studied in [29].

###### Remark 7

A class of RF which lie between Gaussian and RG RFs and are quite popular in
applied sciences are shot-noise RFs having a representation

$X(\boldsymbol{t}):=\sum_{j=1}^{\infty}W_{j}(\boldsymbol{t}-\boldsymbol{u}_{j}),\qquad\boldsymbol{t}\in\mathbb{R}^{d}$
(4.19)

w.r.t. the same Poisson point process $\\{\boldsymbol{u}_{j}\\}$ as in (1.3),
where $W_{j}=\\{W_{j}(\boldsymbol{t});\boldsymbol{t}\in\mathbb{R}^{d}\\}$ are
i.i.d. copies of (generic) pulse RF
$W=\\{W(\boldsymbol{t});\boldsymbol{t}\in\mathbb{R}^{d}\\}$, all independent
of $\\{\boldsymbol{u}_{j}\\}$. We see that (4.19) encompasses (1.3) which
correspond to $W(\boldsymbol{t})=\mathbb{I}(\boldsymbol{t}\in
R^{1/d}\Xi^{0})$. Assuming that the trajectories of $W$ belong to
$L^{1}(\mathbb{R}^{d})$ a.s., (4.19) can be written as the Poisson stochastic
integral

$X(\boldsymbol{t})=\int_{\mathbb{R}^{d}\times
L^{1}(\mathbb{R}^{d})}w(\boldsymbol{t}-\boldsymbol{u}){\cal
N}(\mathrm{d}\boldsymbol{u},\mathrm{d}w),\qquad\boldsymbol{t}\in\mathbb{R}^{d}$
(4.20)

w.r.t. to Poisson random measure ${\cal N}(\mathrm{d}u,\mathrm{d}w)$ with mean
$\mathrm{E}{\cal
N}(\mathrm{d}\boldsymbol{u},\mathrm{d}w)=\mathrm{d}\boldsymbol{u}\mathrm{P}(W(\cdot)\in\mathrm{d}w),(\boldsymbol{u},w)\in\mathbb{R}^{d}\times
L^{1}(\mathbb{R}^{d})$. The integral in (4.20) is a well-defined and
stationary RF with finite variance provided
$\int_{\mathbb{R}^{d}}(\mathrm{E}|W(\boldsymbol{t})|+\mathrm{E}|W(\boldsymbol{t})|^{2})\mathrm{d}\boldsymbol{t}<\infty$
holds, in which case
$\mathrm{E}X(\boldsymbol{t})=\int_{\mathbb{R}^{d}}\mathrm{E}W(\boldsymbol{t})\mathrm{d}\boldsymbol{t}$
and the covariance ${\rm
Cov}(X(\boldsymbol{0}),X(\boldsymbol{t}))=\int_{\mathbb{R}^{d}}\mathrm{E}W(\boldsymbol{u})W(\boldsymbol{t}+\boldsymbol{u})\mathrm{d}\boldsymbol{u},\
\boldsymbol{t}\in\mathbb{R}^{d}$ may exhibit LRD property under suitable
assumption on the pulse RF $W$. A rather general form of pulse allowing for
LRD (see [1, 12, 43]) is given by

$\displaystyle
W(\boldsymbol{t})=\eta\,a(R^{-1/d}\boldsymbol{t}),\qquad\boldsymbol{t}\in\mathbb{R}^{d},$
(4.21)

where $a(\cdot)\in L^{2}(\mathbb{R}^{d})$ is a deterministic function whereas
$\eta\in\mathbb{R},R>0$ are r.v.s satisfying certain moment conditions. The
corresponding covariance function writes as

$\displaystyle{\rm Cov}(X(\boldsymbol{0}),X(\boldsymbol{t}))$ $\displaystyle=$
$\displaystyle\mathrm{E}[\eta^{2}R\,(a\star
a)\big{(}R^{-1/d}\boldsymbol{t}\big{)}\big{]},\qquad\boldsymbol{t}\in\mathbb{R}^{d},$
(4.22)

where $\star$ denotes convolution. (4.21) may be interpreted as a ‘typical’
‘$a$-pulse’ with random ‘amplitude’ $\eta$ and random ‘frequency’ $R^{-1/d}$.
[12, Lemma 2] provides general conditions on $\eta,R,a\star a$ implying
covariance LRD and regular decay of (4.22) as $\|\boldsymbol{t}\|\to\infty$
for $d=1$ which can be generalized to any $d\geq 1$. Gaussian scaling limits
of shot-noise RFs in (4.19), (4.21) were studied in several papers (see, e.g.,
[12] and the references therein). The recent work [19] discussed scaling
limits and a trichotomy similar to (2.23) for one-dimensional ($d=1$) shot-
noise in (4.19) with rescaled intensity
$M\mathrm{d}\boldsymbol{u}\mathrm{P}(W(\cdot)\in\mathrm{d}w),M=\lambda^{\gamma}$
of the Poisson random measure, for a general class of pulse $W$. An
interesting open problem is to extend the results on nonlinear functionals in
Theorem 2 to shot-noise RF in (4.20)-(4.21).

###### Remark 8

A Cox (or doubly stochastic Poisson) point process
$U=\\{\boldsymbol{u}_{j}\\}$ is a Poisson point process on $\mathbb{R}^{d}$
with random intensity $\zeta(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$, where
$\\{\zeta(\boldsymbol{u});\boldsymbol{u}\in\mathbb{R}^{d}\\}=:\zeta$ is a
nonnegative RF, meaning that the conditional distribution of $U$ given $\zeta$
is Poisson with intensity $\zeta(\boldsymbol{u})\mathrm{d}\boldsymbol{u}$
[37]. Scaling limits of shot-noise RFs driven by Cox process were studied in
[10] and other works. LRD property in such RFs may be due to random intensity
$\zeta$. Extending some results of this sec. to RG models driven by Cox
process seems feasible.

##  5 Application: scaling limits of solutions of Burgers’ equation with
initial RG data

Burgers’ equation with (random) potential initial data is written as

$\displaystyle\partial\vec{v}(t,\boldsymbol{x})/\partial
t+(\vec{v}(t,\boldsymbol{x}),\nabla)\vec{v}(t,\boldsymbol{x})$
$\displaystyle=$
$\displaystyle{\small\frac{1}{2}}\kappa\Delta\vec{v}(t,\boldsymbol{x}),$ (5.1)
$\displaystyle\vec{v}(0,\boldsymbol{x})$ $\displaystyle=$
$\displaystyle-\nabla\xi(\boldsymbol{x}),$

where
$\vec{v}(t,\boldsymbol{x})=(v_{1}(t,\boldsymbol{x}),\cdots,v_{d}(t,\boldsymbol{x})),(t,\boldsymbol{x})\in\mathbb{R}_{+}\times\mathbb{R}^{d}$
is a $\mathbb{R}^{d}$-valued function (velocity field) and
$\xi=\\{\xi(\boldsymbol{x});\boldsymbol{x}\in\mathbb{R}^{d}\\}$ is a scalar
(potential) RF;
$(\vec{v}(t,\boldsymbol{x}),\nabla):=\sum_{i=1}^{d}v_{i}(t,\boldsymbol{x})\partial/\partial
x_{i}$. The parameter $\kappa>0$ is usually called the viscosity parameter.
Burgers’ equation is one of the important equations of mathematical physics.
The solution $\vec{v}(t,\boldsymbol{x})$ with random initial data is a
(vector-valued) RF whose behavior as $t\to\infty$ and/or $\kappa\to 0$
presents considerable physical and mathematical interest and has been
extensively studied in the literature. For a probabilistic approach, we refer
to [1] and the review paper [44]. When $\kappa>0$ is fixed, the natural
parabolic scaling leads to the RF
$\vec{v}_{\lambda}(t,\boldsymbol{x}):=\vec{v}(\lambda^{2}t,\lambda\boldsymbol{x})$
and the problem concerns the limit distribution of RF
$\vec{v}_{\lambda}(t,\boldsymbol{x});(t,\boldsymbol{x})\in\mathbb{R}_{+}\times\mathbb{R}^{d}$
as $\lambda\to\infty$.

The study of Burgers’ equation is facilitated by the Hopf-Cole substitution

$\vec{v}(t,\boldsymbol{x})=-\kappa\nabla\log
u(t,\boldsymbol{x})=-\frac{\kappa\nabla
u(t,\boldsymbol{x})}{u(t,\boldsymbol{x})}$ (5.2)

with a scalar-valued $u(t,\boldsymbol{x})$ satisfying the heat equation
$\partial u(t,\boldsymbol{x})/\partial t={\small\frac{1}{2}}\kappa\Delta
u(t,\boldsymbol{x})$ with the (exponential) initial condition
$u(0+,\boldsymbol{x})=\mathrm{e}^{\xi(\boldsymbol{x})/\kappa},\boldsymbol{x}\in\mathbb{R}^{d}.$
Thus, (5.2) has an explicit representation through the heat kernel
$g(t,\boldsymbol{x},\boldsymbol{y}):=(2\pi\kappa
t)^{-d/2}\exp\\{-\|\boldsymbol{x}-\boldsymbol{y}\|^{2}/2\kappa t\\}$ as the
ratio

$\vec{v}(t,\boldsymbol{x})=-\frac{\kappa\int_{\mathbb{R}^{d}}\nabla
g(t,\boldsymbol{x},\boldsymbol{y})\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}\mathrm{d}\boldsymbol{y}}{\int_{\mathbb{R}^{d}}g(t,\boldsymbol{x},\boldsymbol{y})\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}\mathrm{d}\boldsymbol{y}}.$
(5.3)

Using the fact that $\int_{\mathbb{R}^{\mathrm{d}}}\nabla
g(t,\boldsymbol{x},\boldsymbol{y})\mathrm{d}\boldsymbol{y}=0$, one can replace
$\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}$ in the numerator of (5.3) by
$\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}-\mathrm{E}\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}$,
provided the last expectation is finite and does not depend on
$\boldsymbol{y}$. As a consequence, the rescaled velocity RF writes as

$\vec{v}_{\lambda}(t,\boldsymbol{x})=-\frac{\kappa\int_{\mathbb{R}^{d}}\phi_{t,\boldsymbol{x}}(\boldsymbol{y}/\lambda)(G(\xi(\boldsymbol{y}))-\mathrm{E}G(\xi(\boldsymbol{y})))\mathrm{d}\boldsymbol{y}}{\lambda\int_{\mathbb{R}^{d}}\psi_{t,\boldsymbol{x}}(\boldsymbol{y}/\lambda)G(\xi(\boldsymbol{y}))\mathrm{d}\boldsymbol{y}},$
(5.4)

where $G(x)=\mathrm{e}^{x/\kappa}$ and the integrals in the numerator and
denominator resemble (1.1) with
$\phi(\boldsymbol{y})=\phi_{t,\boldsymbol{x}}(\boldsymbol{y}):=\nabla
g(t,\boldsymbol{x},\boldsymbol{y})$ and
$\phi(\boldsymbol{y})=\psi_{t,\boldsymbol{x}}(\boldsymbol{y}):=g(t,\boldsymbol{x},\boldsymbol{y})$,
respectively. Clearly, for any fixed
$(t,\boldsymbol{x})\in\mathbb{R}_{+}\times\mathbb{R}^{d}$ the above $\phi$’s
belong to $\Phi$ in (2.2).

As mentioned in the Introduction, our aim is the limit distribution of (5.4)
for the initial potential RF

$\xi(\boldsymbol{y}):=M^{-1/2}(X_{M}(\boldsymbol{y})-\mathrm{E}X_{M}(\boldsymbol{y})),\quad\boldsymbol{y}\in\mathbb{R}^{d},$
(5.5)

where $X_{M}$ is the (aggregated) RG model as in Theorem 2 and elsewhere in
this paper, with intensity $M=\lambda^{\gamma}$ increasing with $\lambda$ when
$\gamma>0$. Similarly as in [1] and some other works, we ignore the meaning of
the initial condition in (5.1) for $\xi$ in (5.5) and consider scaling
properties of the Hopf-Cole solution in (5.3) alone. Note
$\mathrm{E}\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}\to\mathrm{E}\mathrm{e}^{Z_{\mu}/\kappa}=\mathrm{e}^{\mu/2\kappa^{2}}\,(M\to\infty),\mathrm{E}[\mathrm{e}^{Z_{\mu}/\kappa}Z_{\mu}]/\mathrm{E}\mathrm{e}^{Z_{\mu}/\kappa}=1/\kappa$
and
$\lambda^{-d}\int_{\mathbb{R}^{d}}g(t,\boldsymbol{x},\boldsymbol{y}/\lambda)\mathrm{e}^{\xi(\boldsymbol{y})/\kappa}\mathrm{d}\boldsymbol{y}\to\mathrm{E}\mathrm{e}^{Z_{\mu}/\kappa}$
in probability by the law of large numbers. The application of Theorem 2 with
$G(x)=\mathrm{e}^{x/\kappa},x\in\mathbb{R}$ and (4.18) (Example 2) yields the
following result.

###### Corollary 3

Let $\vec{v}_{\lambda}(t,\boldsymbol{x})$ be as in (5.4), (5.5), with
$X_{M},M$ satisfying the conditions of Theorem 2. Then, as $\lambda\to\infty$,
for any $\gamma>0$

$\displaystyle\lambda^{1+d+\frac{\gamma}{2}-H(\gamma)}\vec{v}_{\lambda}(t,\boldsymbol{x})$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{\longrightarrow}}\\\ \end{array}$
$\displaystyle\begin{cases}B_{\alpha}(\nabla
g(t,\boldsymbol{x},\cdot)),&\gamma>d(\alpha-1),\\\ L_{\alpha}(\nabla
g(t,\boldsymbol{x},\cdot)),&\gamma<d(\alpha-1),\\\ J_{\alpha}(\nabla
g(t,\boldsymbol{x},\cdot)),&\gamma=d(\alpha-1),\end{cases}$ (5.7)

where $H(\gamma)$ and the limit RFs are the same as in (2.14)

Let us remark that convergence to $\alpha$-stable limit in (5.7) holds also
for $\gamma=0$ or $\xi(\boldsymbol{y})$ in (5.5) replaced by
$\xi(\boldsymbol{y})=X(\boldsymbol{y})$ in (1.4). From Proposition 2 we
conclude the following result.

###### Corollary 4

Let $\vec{v}_{\lambda}(t,\boldsymbol{x})$ be as in (5.4) with
$\xi(\boldsymbol{y})=X(\boldsymbol{y})$ given in (1.4). Then, as
$\lambda\to\infty$

$\displaystyle\lambda^{1+d-\frac{d}{\alpha}}\vec{v}_{\lambda}(t,\boldsymbol{x})$
$\displaystyle\begin{array}[t]{c}\stackrel{{\scriptstyle\rm
fdd}}{{\longrightarrow}}\\\ \end{array}$
$\displaystyle\kappa(\mathrm{e}^{1/\kappa}-1)L_{\alpha}(\nabla
g(t,\boldsymbol{x},\cdot)),$ (5.9)

where $L_{\alpha}$ is defined in (2)-(2.12).

We remark that in dimension $d=1$, a similar result to (5.9) was proved in
[40, Thm.1.1(iii)] for initial (potential) data $\xi(y),y\in\mathbb{R}$ given
by a piecewise-constant renewal-reward process with renewal distribution
having $\alpha$-tail with $\alpha\in(1,2)$.

## Acknowledgement

The author is grateful to two anonymous referees for useful and constructive
comments.

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