Limit theorems for filtered long-range dependent random fields

TL;DR

This paper explores the limit distributions of functionals of filtered long-range dependent Gaussian random fields, revealing non-Gaussian limits and the influence of observation window shapes on the Hurst parameter.

Contribution

It extends existing results by establishing convergence for Hurst parameters in (0, 0.5) and analyzing the effect of observation window shapes on the Hurst parameter.

Findings

Limit distributions are non-Gaussian.

Convergence established for H in (0, 0.5).

Hurst parameter depends on observation window shape.

Abstract

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter $H\in(\frac{1}{2},1)$. In this work we also obtain convergence for the case $H\in(0,\frac{1}{2})$ and show how the Hurst parameter $H$ can depend on the shape of the observation windows. Various examples are presented.

Equations

$$X_r^{(d)}(t)=\frac{1}{r^H\mathcal{L}(r)}\sum _{s=1}^{[rt]}V(s)\stackrel{D}{\to }{B}_H(t),t>0,\text{as}r\to \infty ,$$

$$X_r^{(c)}(t)=\frac{1}{r^H\mathcal{L}(r)}{\int }_0^{rt}V(s)ds\stackrel{D}{\to }{B}_H(t),t>0,\text{as}r\to \infty ,$$

$$V_{\kappa }(t)=\sum _{j\in \mathbb{Z}}{G}_{t-j}{H}_{\kappa }({\xi }_j),$$

$$\frac{1}{A_r}\sum _{s=1}^{[rt]}{V}_{\kappa }(s)\stackrel{D}{\to }{Y}_{\kappa }(t),t>0,\text{as}r\to \infty ,$$

$$d_r^{-1}{\int }_{\Delta (r{t}^{1/n})}V(x)dx,t\in [0,1],\text{as}r\to \infty ,$$

$$V(x):={\int }_{{\mathbb{R}}^n}G(\Vert y-x\Vert )S(\xi (y))dy,$$

$$\underset{r\to \infty }{\lim }\frac{\mathcal{L}(tr)}{\mathcal{L}(r)}=1.$$

$$\mathcal{L}(r)=\exp \left({\zeta }_1(r)+{\int }_C^r\frac{{\zeta }_2(u)}{u}du\right),$$

$$\underset{r\to \infty }{\lim }\frac{g(tr)}{g(r)}=t^{\tau }.$$

$$\underset{a\le t\le x}{\sup }g(t)\sim g(x)\mathrm{and}\underset{t\ge x}{\inf }g(t)\sim g(x),x\to \infty .$$

$$\underset{t\ge x}{\sup }g(t)\sim g(x)\mathrm{and}\underset{a\le t\le x}{\inf }g(t)\sim g(x),x\to \infty .$$

$$H_m(x):=(-1{)}^m\exp \left(\frac{x^2}{2}\right)\frac{d^m}{d{x}^m}\exp \left(-\frac{x^2}{2}\right).$$

$$H_0(x)=1,H_1(x)=x,H_2(x)=x^2-1,H_3(x)=x^3-3x.$$

$$S(\omega )=\sum _{j=0}^{\infty }\frac{C_j{H}_j(\omega )}{j!},C_j:={\int }_{\mathbb{R}}S(\omega )H_j(\omega )\varphi (\omega )d\omega .$$

$$\sum _{j=0}^{\infty }\frac{C_j^2}{j!}={\int }_{\mathbb{R}}{S}^2(\omega )\varphi (\omega )d\omega .$$

$$B\left(r\right):=\mathbb{E}\left(\xi (0)\xi (x)\right),x\in {\mathbb{R}}^n,r=\Vert x\Vert .$$

$$B\left(r\right)={\int }_0^{\infty }{Y}_n\left(ru\right)d\Phi \left(u\right),$$

$$Y_n\left(u\right):=2^{(n-2)/2}\Gamma \left(\frac{n}{2}\right)J_{(n-2)/2}(u)u^{(2-n)/2},u⩾0,$$

$$u^{n-1}f(u)\in L_1([0,\infty )),\Phi (u)=2{\pi }^{n/2}/\Gamma (n/2){\int }_0^u{z}^{n-1}f(z)dz,$$

$$\xi \left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{i⟨\lambda ,x⟩}\sqrt{f(\Vert \lambda \Vert )}W(d\lambda ),$$

$$\mathbb{E}\left(H_{m_1}(\xi (x))H_{m_2}(\xi (y))\right)={\delta }_{m_1}^{m_2}{m}_1!B^{m_1}(\Vert x-y\Vert ),x,y\in {\mathbb{R}}^n,$$

$$B(x)=\mathbb{E}\left(\xi \left(0\right)\xi \left(x\right)\right)=\Vert x{\Vert }^{-\alpha }{\mathcal{L}}_0\left(\Vert x\Vert \right),\alpha >0,$$

$$K_{\kappa }:={\int }_{△(r)}S\left(\xi \left(x\right)\right)dx\mathrm{and}{K}_{r,\kappa }:=\frac{C_{\kappa }}{\kappa !}{\int }_{△(r)}{H}_{\kappa }\left(\xi \left(x\right)\right)dx,$$

$$\frac{K_r}{\sqrt{Var{K}_r}}and\frac{K_{r,\kappa }}{\sqrt{Var{K}_{r,\kappa }}},$$

$$f\left(\Vert \lambda \Vert \right)=c_1\left(n,\alpha \right)\Vert \lambda {\Vert }^{\alpha -n}\mathcal{L}\left(\frac{1}{\Vert \lambda \Vert }\right),$$

$$K_{\Delta }\left(x\right):={\int }_{\Delta }{e}^{i⟨u,x⟩}du,x\in {\mathbb{R}}^n.$$

$$X_{r,\kappa }(\Delta ):=r^{\kappa \alpha /2-n}{\mathcal{L}}^{-\kappa /2}(r){\int }_{\Delta (r)}{H}_{\kappa }\left(\xi (x)\right)dx$$

$$X_{\kappa }(\Delta ):=c_1^{\kappa /2}(n,\alpha ){\int }_{{\mathbb{R}}^{n\kappa }}^{\prime }{K}_{\Delta }\left({\lambda }_1+\cdots +{\lambda }_{\kappa }\right)\frac{W(d{\lambda }_1)\cdots W(d{\lambda }_{\kappa })}{\Vert {\lambda }_1{\Vert }^{(n-\alpha )/2}\cdots \Vert {\lambda }_{\kappa }{\Vert }^{(n-\alpha )/2}}.$$

$$\tilde{C}:=\kappa !\underset{r\to \infty }{\lim }{\int }_{\Delta }{\int }_{\Delta }\frac{\vartheta (r\Vert x\Vert )\vartheta (r\Vert y\Vert )dxdy}{{\vartheta }^2(r)\Vert x-y{\Vert }^{\alpha \kappa }}\in \left(0,\infty \right),$$

$$\displaystyle\int_{\mathbb{R}^{n\kappa}}\prod_{j=1}^{\kappa}\|\lambda_{j}\|^{\alpha-n}\bigg{|}\int_{\Delta(t^{1/n})}e^{i\langle\lambda_{1}+\dots+\lambda_{\kappa},x\rangle}\bar{\vartheta}(x)dx\bigg{|}^{2}\prod_{j=1}^{\kappa}d\lambda_{j}<\infty$$

Full text

1 Introduction

Over the last four decades, several studies dealt with various functionals of random fields and their asymptotic behaviour anh2017rate , anh2015rate , bai2013multivariate , dobrushin1979non , doukhan2002theory , taqqu1979convergence , taqqu1975weak , ivanov2008semiparametric , leonenko2014sojourn , kratz2017central . These functionals play an important role in various fields, such as physics, cosmology, telecommunications, just to name a few. In particular, asymptotic results were obtained either for integrals or additives functionals of random fields under long-range dependence, see leonenko2006weak , weak2017alodat , olenko2010limit , olenko2013limit , doukhan2002theory , nourdin2014central , leonenko2017rosenblatt and the references therein.

It is well known that functionals of Gaussian random fields with long-range dependence can have non-Gaussian asymptotics and require normalising factors different from those in central limit theorems. These limit processes are known as Hermite or Hermite-Rosenblatt processes. The first result in this direction was obtained in rosenblatt1961independence where quadratic functionals of long-range dependent stationary Gaussian sequences were investigated. The pioneering results in the asymptotic theory of non-linear functionals of long-range dependent Gaussian processes and sequences can be found in taqqu1979convergence , taqqu1975weak , dobrushin1979non , rosenblatt1981limit , taqqu1978representation . This line of studies attracted much attention, for example, in pakkanen2016functional it was shown that the limiting distribution of generalised variations of a long-range dependent fractional Brownian sheet is a fractional Brownian sheet that is independent and different from the original one. Some statistical properties of the Rosenblatt distribution, as well as its expansion in terms of shifted chi-squared distributions were studied in veillette2013 . The Lévy-Khintchine formula and asymptotic properties of the Lévy measure, were also addressed in leonenko2017rosenblatt . Some weighted functionals for long-range dependent random fields were considered and limit theorems were investigated in a number of papers, including olenko2013limit , ivanov2013limit , ivanov1989statistical .

Linear stochastic processes and random fields obtained as outputs of filters are popular models in various applications, see jazwinski1970stochastic , wiener1949extrapolation , kallianpur2013stochastic , alomari2018estimation . In engineering practice it is often assumed that a narrow band-pass filter applied to a stationary random input yields an approximately normally distributed output. Of course, such results are not true in general, especially when the stationary input has some singularity in the spectrum and the linear filtration is replaced by a non-linear one.

We recall the classical central-limit type theorem by Davydov davydov1970invariance for discrete time linear stochastic processes.

Theorem 1.1

davydov1970invariance * Let $V(t)=\sum_{j\in\mathbb{Z}}G_{t-j}\xi_{j},\ t\in\mathbb{Z}$, where ${\xi_{j}}$ is a sequence of i.i.d random variables with zero mean and finite variance (the $\{\xi_{j}\}$ are not necessarily Gaussian). Suppose that ${G_{j}}$ is a real-valued sequence satisfying $\sum_{j\in\mathbb{Z}}G_{j}^{2}<\infty$ and let $X_{r}^{(d)}:=\sum_{t=1}^{r}V(t)$. If $VarX_{r}^{(d)}=r^{2H}\mathcal{L}^{2}(r)$ as $r\to\infty$, where $H\in(0,1)$ and the function $\mathcal{L}(\cdot)$ is a slowly varying at infinity, then*

[TABLE]

in the sense of convergence of finite-dimensional distributions, where $B_{H}(t)$, $t>0$, is the fractional Brownian motion with zero mean and the covariance function $B_{H}(t,s)=\frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right),\ t,s>0,\ 0<H<1.$

One can obtain an analogous result for the case of continuous time.

Theorem 1.2

Let $V(t)=\int_{\mathbb{R}}G(t-s)\xi(s)ds,\ t\in\mathbb{R}$, be a linear filtered process, where $\xi(t),\ t\in\mathbb{R}$, be a mean-square continuous stationary in the wide sense process with zero mean and finite variance. Suppose that $G(t),\ t\in\mathbb{R}$, is a non-random function, such that $\int_{\mathbb{R}}G^{2}(t)dt<\infty$. Let $X_{r}^{(c)}:=\int_{0}^{r}V(s)ds$. If $VarX_{r}^{(c)}=r^{2H}\mathcal{L}^{2}(r),$ as $r\to\infty$, where $H\in(0,1)$ and $\mathcal{L}(\cdot)$ is slowly varying at infinity, then

[TABLE]

in a sense of convergence of finite-dimensional distributions.

The equivalence of the statements for the discrete and continuous time follows from the results in Leonenko and Taufer leonenko2006weak and Alodat and Olenko weak2017alodat .

It was Rossenblatt Rosenblatt1979 (see also Major major1981limit , Taqqu doukhan2002theory ) who first proved that for a discrete-time Gaussian stochastic process $\{\xi_{j},\ j\in\mathbb{Z}\}$, with zero mean and long-range dependence and the $\kappa$-th Hermite polynomials $H_{\kappa}$, the non-linear filtered process

[TABLE]

satisfies the non-central limit theorem, that is for some normalising $A_{r}$ it holds

[TABLE]

where $Y_{\kappa}(t)$ is a self-similar process with the Hurst parameter $H\in(0,1)$ (non-Gaussian, if $\kappa\geq 2$).

The limit processes $Y_{\kappa}(t),\ t>0$, are given in terms of $\kappa$-fold Wiener-Itô stochastic integrals, and are the fractional Brownian motions with the Hurst parameter $H\in(0,1)$ if $\kappa=1$.

The aim of this paper is to give an extension of the results of RossenblattRosenblatt1979 , Major major1981limit , Taqqu doukhan2002theory for the case of random fields. Motivated by the theory of renormalisation and homogenisation of solutions of randomly initialised partial differential equations (PDE) and fractional partial differential equations (FPDE) (see, e.g. albeverio1994stratified , leonenko1998exact , liu2010scaling , leonenko1998scaling ), we study the asymptotic behaviour of integrals of the form

[TABLE]

where $V(x),\ x\in\mathbb{R}^{n}$, is a random field, $\Delta\subset\mathbb{R}^{n}$ is an observation window and $d_{r}$ is a normalising factor. The case when the limit process is self-similar with parameter $H\in(0,1)$ is considered.

The parameter $H$ plays an important role in analysing stochastic processes and can be used for their classification. In particular, stochastic processes can be classified according to the range of $H$ to the Brownian motion $(H=0.5)$, a short-memory anti-persistent stochastic process $(H\in(0,0.5))$ and a long-memory stochastic process ($H\in(0.5,1)$). These three cases correspond to the three types of behaviour called $\frac{1}{f}$ noise, ultraviolet and infrared catastrophes by Mandelbrot and Taqqu taqqu1981self . The literature shows a variety of limit theorems with asymptotics given by non-Gaussian self-similar processes that exhibit non-negative auto-correlation structures with parameter $H\in(0.5,1)$, see taqqu1975weak , taqqu1979convergence , ivanov1989statistical , leonenko1999limit , olenko2013limit , olenko2010limit and references therein. However, there are only few results where asymptotic processes have $H\in(0,0.5)$. In the case $H<0.5$ processes exhibit a negative dependence structure, which is useful in applied modelling of switching between high and low values. Also, such processes have interesting theoretical stochastic properties. For example, in this case the covariance is the Green function of a Markov process and the squared process is infinitely divisible, which is not true for the case $H>0.5$, seeeisenbaum2005squared , eisenbaum2006characterization .

The example of a non-Gaussian self-similar process with $H\in(0,0.5)$ was given by Rossenblatt Rosenblatt1979 where the asymptotic of quadratic functions of a long-range Gaussian stationary sequence was investigated. The result was generalised in major1981limit for sums of non-linear functionals of Gaussian sequences. In this paper we extend these results in several directions for more general conditions and derive limit theorems for functionals of filtered random fields defined as the convolution

[TABLE]

where $G(\cdot)$, $S(\cdot)$ are non-random functions and $\xi(\cdot)$ is a long-range dependent random field.

In the limit theorems obtained in this paper the asymptotic processes have the self-similar parameters $H\in\left(\gamma(\Delta),1\right)$, where $\gamma(\Delta)\geq 0$ depends on the geometry of the set $\Delta\subset\mathbb{R}^{n}$. In the one-dimensional case $d=1$, $\gamma(\Delta)=0$ which coincides with the known results in the literature.

The rest of the article is organised as follows. In Section 2 we outline the necessary background. In Section 3 we introduce assumptions and give auxiliary results from the spectral and correlation theory of random fields. In Section 4 we present main results on the asymptotic behaviour of functionals of filtered random fields. Examples are presented in Section 5.

2 Notations

This section gives main definitions and notations that are used in this paper.

In what follows $|\cdot|$ and $\|\cdot\|$ are used for the Lebesque measure and the Euclidean distance in $\mathbb{R}^{n},\ n\geq 1$, respectively. The symbols $C$, $\epsilon$ and $\delta$ (with subscripts) will be used to denote constants that are not important for our discussion. Moreover, the same symbol may be used for different constants appearing in the same proof.

Definition 1

A real-valued function $h:[0,\infty)\rightarrow\mathbb{R}$ is homogeneous of degree $\beta$ if $h(ax)=a^{\beta}h(x)$ for all $a,\ x>0$.

Definition 2

bingham1987regular A measurable function $\mathcal{L}:(0,\infty)\rightarrow(0,\infty)$ is slowly varying at infinity if for all $t>0$,

[TABLE]

By the representation theorem [bingham1987regular, , Theorem 1.3.1], there exists $C>0$ such that for all $r\geq C$ the function $\mathcal{L}(\cdot)$ can be written in the form

[TABLE]

where $\zeta_{1}(\cdot)$ and $\zeta_{2}(\cdot)$ are such measurable and bounded functions that $\zeta_{2}(r)\rightarrow 0$ and $\zeta_{1}(r)\rightarrow C_{0}$, $\left(C_{0}<\infty\right)$, when $r\rightarrow\infty$.

If $\mathcal{L}(\cdot)$ varies slowly, then $r^{a}\mathcal{L}(r)\rightarrow\infty$, and $r^{-a}\mathcal{L}(r)\rightarrow 0$ for an arbitrary $a>0$ when $r\rightarrow\infty$, see Proposition 1.3.6 bingham1987regular .

Definition 3

bingham1987regular A measurable function $g:(0,\infty)\rightarrow(0,\infty)$ is regularly varying at infinity, denoted $g(\cdot)\in R_{\tau}$, if there exists $\tau$ such that, for all $t>0$, it holds that

[TABLE]

Theorem 2.1

[bingham1987regular, , Theorem 1.5.3]* Let $g(\cdot)\in R_{\tau}$, and choose $a\geq 0$ so that $g$ is locally bounded on $[a,\infty)$. If $\tau>0$ then*

[TABLE]

If $\tau<0$ then

[TABLE]

Definition 4

The Hermite polynomials $H_{m}(x),\ m\geq 0$, are given by

[TABLE]

The first few Hermite polynomials are

[TABLE]

The Hermite polynomials $H_{m}(x),\ m\geq 0$, form a complete orthogonal system in the Hilbert space $L_{2}\left(\mathbb{R},\phi(\omega)d\omega\right)=\left\{S:\int_{\mathbb{R}}S^{2}(\omega)\phi(\omega)d\omega<\infty\right\}$, where $\phi(\omega)$ is the probability density function of the standard normal distribution.

An arbitrary function $S(\omega)\in L_{2}\left(\mathbb{R},\phi(\omega)d\omega\right)$ possesses the mean-square convergent expansion

[TABLE]

By Parseval’s identity

[TABLE]

Definition 5

taqqu1975weak Let $S(\omega)\in L_{2}\left(\mathbb{R},\phi(\omega)d\omega\right)$ and there exists an integer $\kappa\geqslant 1$, such that $C_{j}=0$ for all $0<j\leq\kappa-1$, but $C_{\kappa}\neq 0$. Then $\kappa$ is called the Hermite rank of $S(\cdot)$ and is denoted by $HrankS(\cdot).$

It is assumed that all random variables are defined on a fixed probability space $\left(\Omega,\mathfrak{F},\mathbb{P}\right)$. We consider a measurable mean-square continuous zero-mean homogeneous isotropic real-valued random field $\xi\left(x\right),\ x\in\mathbb{R}^{n}$, with the covariance function

[TABLE]

It is well known that there exists a bounded non-decreasing function $\Phi\left(u\right)$, $u\geq 0$, (see ivanov1989statistical , yadrenko1983spectral ) such that

[TABLE]

where the function $Y_{n}\left(\cdot\right),\ n\geq 1,$ is defined by

[TABLE]

where $J_{(n-2)/2}(\cdot)$ is the Bessel function of the first kind of order $(n-2)/2$, see leonenko1999limit , yadrenko1983spectral . The function $\Phi\left(\cdot\right)$ is called the isotropic spectral measure of the random field $\xi\left(x\right),\ x\in\mathbb{R}^{n}$.

Definition 6

If there exists a function $f(u),\ u\in[0,\infty)$, such that

[TABLE]

then the function $f(\cdot)$ is called the isotropic spectral density of the field $\xi\left(x\right)$.

The field $\xi\left(x\right)$ with an absolutely continuous spectrum has the following isonormal spectral representation

[TABLE]

where $W(\cdot)$ is the complex Gaussian white noise random measure on $\mathbb{R}^{n}$, see ivanov1989statistical , leonenko1999limit , yadrenko1983spectral .

Note, that by (2.1.8) leonenko1999limit we get $\mathbb{E}\left(H_{m}(\xi(x))\right)=0$ and

[TABLE]

where $\delta_{m_{1}}^{m_{2}}$ is the Kronecker delta function.

Definition 7

A random process $X(t),\ t>0$, is called self-similar with parameter $H>0$, if for any $a>0$ it holds $X(at)\stackrel{{\scriptstyle D}}{{=}}a^{H}X(t)$.

If $X(t),\ t>0$, is a self-similar process with parameter $H>0$ such that $\mathbb{E}(X(t))=0$ and $\mathbb{E}(X^{2}(t))<\infty$, then $B(at,as)=a^{2H}B(t,s)$, see leonenko1999limit .

3 Assumptions and auxiliary results

This section introduces assumptions and results from the spectral and correlation theory of random fields.

Assumption 1

Let $\xi(x),\ x\in\mathbb{R}^{n}$, be a homogeneous isotropic Gaussian random field with $\mathbb{E}\xi(x)=0$ and the covariance function $B(x)$, such that $B(0)=1$ and

[TABLE]

where $\mathcal{L}_{0}\left(\|\cdot\|\right)$ is a function slowly varying at infinity.

If $\alpha\in\left(0,n\right)$, then the covariance function $B(x)$ satisfying Assumption 1 is not integrable, which corresponds to the long-range dependence caseanh2015rate .

The notation $\Delta\subset\mathbb{R}^{n}$ will be used to denote a Jordan-measurable compact bounded set, such that $|\Delta|>0$, and $\Delta$ contains the origin in its interior. Let $\Delta(r),\ r>0$, be the homothetic image of the set $\Delta$, with the centre of homothety at the origin and the coefficient $r>0$, that is $|\Delta(r)|=r^{n}|\Delta|$ and $\Delta=\Delta(1)$.

Let $S(\omega)\in L_{2}\left(\mathbb{R},\phi(\omega)d\omega\right)$ and denote the random variables $K_{\kappa}$ and $K_{r,\kappa}$ by

[TABLE]

where $C_{\kappa}$ is given by (1).

Theorem 3.1

leonenko2014sojourn * Suppose that $\xi\left(x\right),\ x\in\mathbb{R}^{n}$, satisfies Assumption 1 and $HrankS(\cdot)=\kappa\geq 1$. If a limit distribution exists for at least one of the random variables*

[TABLE]

then the limit distribution of the other random variable also exists, and the limit distributions coincide when $r\rightarrow\infty$.

By Theorem 3.1 it is enough to study $K_{r,\kappa}$ to get asymptotic distributions of $K_{\kappa}$. Therefore, we restrict our attention only to $K_{r,\kappa}$.

Assumption 2

The random field $\xi\left(x\right),\ x\in\mathbb{R}^{n}$, has the isotropic spectral density

[TABLE]

where $\alpha\in(0,n),\ c_{1}\left(n,\alpha\right):=\Gamma\left(\frac{n-\alpha}{2}\right)/2^{\alpha}\pi^{n/2}\Gamma\left(\frac{\alpha}{2}\right),$ and $\mathcal{L}(\|\cdot\|)\sim\mathcal{L}_{0}(\|\cdot\|)$ is a locally bounded function which is slowly varying at infinity.

One can find more details on relations between Assumptions 1 and 2 in anh2017rate , anh2015rate .

The function $K_{\Delta}\left(x\right)$ will be used to denote the Fourier transform of the indicator function of the set $\Delta$, i.e.

[TABLE]

Theorem 3.2

leonenko2014sojourn * Let $\xi\left(x\right),\ x\in\mathbb{R}^{n}$, be a homogeneous isotropic Gaussian random field. If Assumptions 1 and 2 hold, then for $r\rightarrow\infty$ the random variables*

[TABLE]

converge weakly to

[TABLE]

Here $\int_{\mathbb{R}^{n\kappa}}^{\prime}$ denotes the multiple Wiener-Itô integral with respect to a Gaussian white noise measure, where the diagonal hyperplanes $\lambda_{i}=\pm\lambda_{j},\ i,j=1,\dots,\kappa,\ i\neq j$, are excluded from the domain of integration.

Assumption 3

ivanov1989statistical * Let $\vartheta(x)=\vartheta(\|x\|)$ be a radial continuous function positive for $\|x\|>0$ and such that there exists*

[TABLE]

where $\alpha\in\left(0,n/\kappa\right)$.

Let $u(\|\lambda\|):=c_{1}\left(n,\alpha\right)\mathcal{L}\left(\dfrac{1}{\|\lambda\|}\right)$, where $\mathcal{L}(\cdot)$ is from Assumption 2. In leonenko1999limit and Section 10.2 ivanov1989statistical the case when the function $u(\|\lambda\|)$ is continuous in a neighborhood of zero, bounded on $(0,\infty)$ and $u(0)\neq 0$, was studied. It was assumed that there is a function $\bar{\vartheta}(\|x\|)$ such that

[TABLE]

and

[TABLE]

for all $t\in[0,1]$.

Under these assumptions the following result was obtained.

Theorem 3.3

ivanov1989statistical * If Assumption 3 holds, then the finite-dimensional distributions of the random processes*

[TABLE]

converge weakly to finite-dimensional distributions of the processes

[TABLE]

as $r\to\infty$, where $\alpha\in\left(0,\min\left(\frac{n}{\kappa},\frac{n+1}{2}\right)\right)$ and

[TABLE]

4 Limit theorems for functionals of filtered fields

This section derives the generalisation of Theorem 3.3 when the integrand $\vartheta(\cdot)H_{\kappa}\left(\cdot\right)$ in (4) is replaced by a filtered random field.

Assumption 4

Let $h:[0,\infty)\rightarrow\mathbb{R}$ be a measurable real-valued homogeneous function of degree $\beta$ and $g:[0,\infty)\rightarrow\mathbb{R}$ be a bounded uniformly continuous function such that $g(0)\neq 0$ in some neighberhood of zero and $\int_{\mathbb{R}^{n}}h^{2}(\|u\|)g^{2}(\|u\|)du<\infty$.

We define the filtered random field $V(x),\ x\in\mathbb{R}^{n}$, as

[TABLE]

where

[TABLE]

is the Fourier transform of $h(\cdot)g(\cdot)$.

Remark 1

Note, that from the isonormal spectral representation (2) and the Itô formula

[TABLE]

it follows that

[TABLE]

where $\hat{G}(\cdot)$ is the Fourier transform of the function $G(\cdot)$ that is defined by (6) and the stochastic Fubini’s theorem [peccati2011wiener, , Theorem 5.13.1] was used to interchange the order of integration.

By (6) and Assumption 4 the isonormal spectral representation of $V(x)$ is

[TABLE]

Therefore, it follows that the covariance of $V(x)$ is

[TABLE]

Remark 2

By the homogeneity of $h(\cdot)$ and Lemma 3 in leonenko2014sojourn it holds

[TABLE]

for $\alpha\in\left(0,n-2\beta\right)$ and $\beta<n/2$.

Lemma 1

*If $\tau_{1},\dots,\tau_{\kappa},\ \kappa\geq 1$, are positive constants such that it holds

$\sum_{i=1}^{\kappa}\tau_{i}<n-2\beta$ and $\beta<n/2$, then*

[TABLE]

Proof

For $\kappa=1$ we have $\tau_{1}\in(0,n-2\beta)$ and by Remark 2 we get the statement of the Lemma.

For $\kappa>1$, let us use the change of variables $\tilde{\lambda}_{\kappa-1}=\lambda_{\kappa-1}/\|u\|$, where $u=\lambda_{\kappa}+\lambda_{\kappa-1}$. Then, by applying the recursive estimation routine we get

[TABLE]

[TABLE]

Note, that the second integrand in (4) is unbounded at $\|\tilde{\lambda}_{\kappa-1}\|=0$ and $\tilde{\lambda}_{\kappa-1}=u/\|u\|$ (in this case $\|\tilde{\lambda}_{\kappa-1}\|=1$). If we split $\mathbb{R}^{n}$ into the regions $A_{1}:=\{\tilde{\lambda}_{\kappa-1}\in\mathbb{R}^{n}:\|\tilde{\lambda}_{\kappa-1}\|<\frac{1}{2}\}$, $A_{2}:=\{\tilde{\lambda}_{\kappa-1}\in\mathbb{R}^{n}:\frac{1}{2}\leq\|\tilde{\lambda}_{\kappa-1}\|<\frac{3}{2}\}$, and $A_{3}:=\{\tilde{\lambda}_{\kappa-1}\in\mathbb{R}^{n}:\|\tilde{\lambda}_{\kappa-1}\|\geq\frac{3}{2}\}$ we get

[TABLE]

where $A_{2}-\dfrac{u}{\|u\|}=\{\lambda\in\mathbb{R}^{n}:\lambda+\dfrac{u}{\|u\|}\in A_{2}\}\subset v_{n}\left(\frac{5}{2}\right)$, $v_{n}\left(r\right)$ is a $n$-dimensional ball with center [math] and radius $r$.

Hence, by (4) and Remark 2

[TABLE]

which completes the proof.∎

Lemma 2

The following integral is finite

[TABLE]

Proof

As $f(\cdot)$ is an isotropic spectral density we can rewrite $\mathcal{J}_{\kappa}$ as

[TABLE]

Note that $|\hat{G}|^{2}(\cdot)\in L_{1}\left(\mathbb{R}^{n}\right)$ and $f(\cdot)\in L_{1}\left(\mathbb{R}^{n}\right)$. Hence, by Young’s theorem bogachev2007measure it follows that \hat{G}_{1}^{2}(\cdot)=\left(\big{|}\hat{G}\big{|}^{2}*f\right)(\cdot)\in L_{1}\left(\mathbb{R}^{n}\right). Therefore, using convolutions as in (4) we obtain

[TABLE]

where $\hat{G}_{j+1}^{2}(\cdot):=\left(\hat{G}_{j}^{2}*f\right)(\cdot)\in L_{1}\left(\mathbb{R}^{n}\right)$ by Young’s theorem and recursive steps.∎

Now we proceed to the main result.

Theorem 4.1

Let $\xi\left(x\right),\ x\in\mathbb{R}^{n}$, be a random field satisfying Assumptions 1, 2 and functions $g(\cdot)$ and $h(\cdot)$ satisfy Assumption 4. Then, for $r\rightarrow+\infty$ the finite-dimensional distributions of

[TABLE]

converge weakly to the finite-dimensional distributions of

[TABLE]

where $\alpha\in\left(0,\frac{n-2\beta}{\kappa}\right)$ and $\beta<\frac{n}{2}$.

Remark 3

By the representation (1) in Remark 1 of the covariance function of $V(x)$ we obtain

[TABLE]

In particular, the variance of $X_{r,\kappa}(t)$ is equal

[TABLE]

Similarly, we get

[TABLE]

and

[TABLE]

Remark 4

Note, that for $a>0$ we have

[TABLE]

Using the transformation $\tilde{\lambda}_{j}=a^{\frac{1}{n}}\lambda_{j},\ j=1,\dots,\kappa$, and the self-similarity of the Gaussian white noise we get

[TABLE]

Thus, the random process $X_{\kappa}(t)$ is a self-similar with the Hurst parameter $H=1-\frac{\kappa\alpha}{2n}-\frac{\beta}{n}$.

Proof

By (5) the process $X_{r,\kappa}(t)$ admits the following representation

[TABLE]

By (7) we obtain

[TABLE]

By Assumption 2 it follows $\prod_{j=1}^{\kappa}\sqrt{f(\|\lambda_{j}\|)}\in L_{2}\left(\mathbb{R}^{n\kappa}\right)$. By Assumption 4, (6) and Parseval’s theorem $G(\cdot)\in L_{2}(\mathbb{R}^{n})$. So, one can apply the stochastic Fubini’s theorem to interchange the inner integrals in (4), see Theorem 5.13.1 in peccati2011wiener , which results in

[TABLE]

Note, that by Lemma 2 the integrand in (4) belongs to $L_{2}(\mathbb{R}^{n\kappa})$. Then, it follows from the stochastic Fubini’s theorem, and Assumption 4 that

[TABLE]

where

[TABLE]

Note, that $K_{\Delta(rt^{1/n})}(\lambda)=tr^{n}K_{\Delta}\left(\lambda rt^{1/n}\right)$, where $K_{\Delta}(\cdot)$ is given by (3). Therefore,

[TABLE]

Using the transformation $\lambda^{(j)}=r\lambda_{j},\ j=1,\dots,\kappa$, and the self-similarity of the Gaussian white noise we get

[TABLE]

By the isometry property of multiple stochastic integrals

[TABLE]

where

[TABLE]

Note, that by Assumptions 2, 4, and properties of slowly varying functions $Q_{r}(\lambda^{(1)},\dots,\lambda^{(\kappa)})$ pointwise converges to 1, when $r\rightarrow\infty$.

Let us split $\mathbb{R}^{n\kappa}$ into the regions

[TABLE]

where $\mu=\left(\mu_{1},\dots,\mu_{\kappa}\right)\in\{-1,1\}^{\kappa}$ is a binary vector of length $\kappa$. Then we can represent the integral $R_{r}$ as

[TABLE]

If $\left(\lambda^{(1)},\dots,\lambda^{(\kappa)}\right)\in B_{\mu}$ we estimate the integrand as follows

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\delta$ is an arbitrary positive number.

Using the boundedness of the function $g(\cdot)$, we can write

[TABLE]

[TABLE]

[TABLE]

By Theorem 2.1

[TABLE]

and

[TABLE]

Therefore, there exists $r_{0}>0$ such that for all $r\geq r_{0}$ and $\left(\lambda^{(1)},\dots,\lambda^{(\kappa)}\right)\in B_{\mu}$

[TABLE]

[TABLE]

[TABLE]

By Lemma 1, if we choose $\delta\in\left(0,\min\{\alpha,\frac{n-2\beta}{\kappa}-\alpha\}\right)$, the upper bound in (14) is an integrable function on each $B_{\mu}$ and hence on $\mathbb{R}^{n\kappa}$ too. By Lebesque’s dominated convergence theorem $R_{r}\to 0$ as $r\to\infty$, which completes the proof. ∎

5 Examples

The objective of this section is to investigate the Hurst parameter $H$ of the limit process $X_{\kappa}(t)$ in Theorem 4.1. The section provides simple examples where the range $\left(\gamma(\Delta),1\right)$ for $H$ is explicitly specified depending on the observation window $\Delta\subset\mathbb{R}^{n}$.

Recall that $H=1-\frac{\kappa\alpha}{2n}-\frac{\beta}{n}$ and $\mathcal{I}_{1}(\kappa\alpha)$ is defined as

[TABLE]

Example 1

Let $n=1$ and $\Delta$ has the form $\Delta=[-b,a]\subset\mathbb{R}$, where $a,b\geq 0$ and $|a+b|\neq 0$. Using (3) one obtains

[TABLE]

Note, that as $\lambda\to 0$ it holds $|K_{[-b,a]}\left(\lambda\right)|\to b+a<\infty.$

Now, as $\lambda\to\infty$

[TABLE]

Let $\tau_{1}=\tau_{2}=\cdots=\tau_{\kappa}=\alpha$. Then, by (4) $\mathcal{I}_{\kappa}(\alpha,\dots,\alpha)$ can be estimated as

[TABLE]

Note, that the two conditions $1-\kappa\alpha-2\beta<1$ and $3-\kappa\alpha-2\beta>1$ are required to guarantee that, integrals in (1) are finite. The first condition imples $1-\frac{\kappa\alpha}{2}-\beta<1$ and the second one $1-\frac{\kappa\alpha}{2}-\beta>0$. So $\mathcal{I}_{\kappa}(\alpha,\dots,\alpha)<\infty$ if $H\in(0,1)$.

Example 2

Let $\Delta$ be an $n$-dimensional ball of radius 1, i.e. $\Delta=v(1)\subset\mathbb{R}^{n}$. In this case

[TABLE]

Note, that as $\|\lambda\|\to 0$ it holds $|K_{v(1)}\left(\lambda\right)|\leq C<\infty.$

Now, as $\|\lambda\|\to\infty$ we obtain

[TABLE]

Let $\tau_{1}=\tau_{2}=\cdots=\tau_{\kappa}=\alpha$. Then, by (4) $\mathcal{I}_{\kappa}(\alpha,\dots,\alpha)$ can be estimated as

[TABLE]

The two integrals in (2) are finite provided that $1-\kappa\alpha-2\beta<1$ and $n-\kappa\alpha-2\beta+2>1$. It follows that $\frac{1}{2}-\frac{1}{2n}<1-\frac{\kappa\alpha}{2n}-\frac{\beta}{n}<1$, i.e. $H\in(\frac{1}{2}-\frac{1}{2n},1)$, where $n\in\mathbb{N}$. Note, that for $n=1$ the Hurst index $H\in(0,1)$ and one obtains the same result as in Example 1. For $n>1$ we get $\gamma(v(1))=\frac{1}{2}-\frac{1}{2n}<\frac{1}{2}$.

Example 3

Let $n=2$, $\Delta=\Box(1)=[-1,1]^{2}\subset\mathbb{R}^{2}$. In this case

[TABLE]

Note, that $|K_{\Box(1)}\left(\lambda_{1},\lambda_{2}\right)|\leq C$.

When $\min(\lambda_{1},\lambda_{2})>C_{0}>0$ we get

[TABLE]

and if $\lambda_{j}>C_{0}>0,\ \lambda_{i}\leq C_{0},\ i,j\in\{1,2\},\ i\neq j$, then

[TABLE]

Let us split $\mathbb{R}^{2}$ into the regions

[TABLE]

where $C_{0}>0$.

Then $\mathcal{I}_{1}(\kappa\alpha)$ can be written as

[TABLE]

where $\mathcal{I}_{1}^{(j)}(\kappa\alpha):=\int_{A_{j}^{\prime}}\frac{|K_{\Box(1)}\left(\lambda\right)|^{2}d\lambda}{\|\lambda\|^{2-\kappa\alpha-2\beta}},\ j=1,\dots,4$.

We will consider each term in (18) separately. The term $\mathcal{I}_{1}^{(1)}(\cdot)$ can be estimated as

[TABLE]

The last integral is finite provided that $1-\frac{\kappa\alpha}{2}-\beta<1$, i.e. $H<1$.

Using (17) the term $\mathcal{I}_{1}^{(2)}(\cdot)$ can be estimated as

[TABLE]

The last integrals are finite provided that $1-\frac{\kappa\alpha}{2}-\beta<1$ and $3-\frac{\kappa\alpha}{2}-\beta>1$. It follows that $H\in(0,1)$. Similarly, one obtains $\mathcal{I}_{1}^{(3)}(\kappa\alpha)<\infty$ when $H\in(0,1)$.

Now, for the term $\mathcal{I}_{1}^{(4)}(\cdot)$ we obtain

[TABLE]

The last integral is finite provided that $3-\frac{\kappa\alpha}{2}-\beta>1$. It follows that $H>0$.

By combining the above results for (18), one obtains $\mathcal{I}_{1}(\kappa\alpha)<\infty$. Therefore, using $\tau_{1}=\tau_{2}=\cdots=\tau_{\kappa}=\alpha$ and the inequality (4) we obtain that the result of Theorem 4.1 is true when $H\in(0,1)$.

Acknowledgements This research was partially supported under the Australian Research Council’s Discovery Projects (project DP160101366).