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# Limit theorems for filtered long-range dependent random fields

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∈(1/2,1). In this work we also obtain convergence for the case H∈(0,1/2) and show how the Hurst parameter H can depend on the shape of the observation windows. Various examples are presented.

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## 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 , where

is a sequence of i.i.d random variables with zero mean and finite variance

(the are not necessarily Gaussian). Suppose that is a real-valued sequence satisfying and let . If as , where and the function is a slowly varying at infinity, then

 X(d)r(t)=1rHL(r)[rt]∑s=1V(s)D→BH(t), t>0, as r→∞,

in the sense of convergence of finite-dimensional distributions, where , , is the fractional Brownian motion with zero mean and the covariance function

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

###### Theorem 1.2

Let , be a linear filtered process, where , be a mean-square continuous stationary in the wide sense process with zero mean and finite variance. Suppose that , is a non-random function, such that . Let . If as , where and is slowly varying at infinity, then

 X(c)r(t)=1rHL(r)∫rt0V(s)dsD→BH(t), t>0, as r→∞,

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 , with zero mean and long-range dependence and the -th Hermite polynomials , the non-linear filtered process

 Vκ(t)=∑j∈ZGt−jHκ(ξj),

satisfies the non-central limit theorem, that is for some normalising it holds

 1Ar[rt]∑s=1Vκ(s)D→Yκ(t), t>0, as r→∞,

where is a self-similar process with the Hurst parameter (non-Gaussian, if ).

The limit processes , are given in terms of -fold Wiener-Itô stochastic integrals, and are the fractional Brownian motions with the Hurst parameter if .

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

 d−1r∫Δ(rt1/n)V(x)dx,t∈[0,1], as r→∞,

where , is a random field, is an observation window and is a normalising factor. The case when the limit process is self-similar with parameter is considered.

The parameter

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

to the Brownian motion , a short-memory anti-persistent stochastic process and a long-memory stochastic process (). These three cases correspond to the three types of behaviour called 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 , see taqqu1975weak , taqqu1979convergence , ivanov1989statistical , leonenko1999limit , olenko2013limit , olenko2010limit and references therein. However, there are only few results where asymptotic processes have . In the case 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 , seeeisenbaum2005squared , eisenbaum2006characterization .

The example of a non-Gaussian self-similar process with 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

 V(x):=∫RnG(∥y−x∥)S(ξ(y))dy,

where , are non-random functions and is a long-range dependent random field.

In the limit theorems obtained in this paper the asymptotic processes have the self-similar parameters , where depends on the geometry of the set . In the one-dimensional case , 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 and are used for the Lebesque measure and the Euclidean distance in , respectively. The symbols , and (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 is homogeneous of degree if for all .

###### Definition 2

bingham1987regular A measurable function is slowly varying at infinity if for all ,

 limr→∞L(tr)L(r)=1.

By the representation theorem [bingham1987regular, , Theorem 1.3.1], there exists such that for all the function can be written in the form

 L(r)=exp(ζ1(r)+∫rCζ2(u)udu),

where and are such measurable and bounded functions that and , , when .

If varies slowly, then , and for an arbitrary when , see Proposition 1.3.6 bingham1987regular .

###### Definition 3

bingham1987regular A measurable function is regularly varying at infinity, denoted , if there exists such that, for all , it holds that

 limr→∞g(tr)g(r)=tτ.
###### Theorem 2.1

[bingham1987regular, , Theorem 1.5.3] Let , and choose so that is locally bounded on . If then

 supa≤t≤xg(t)∼g(x) and inft≥xg(t)∼g(x),x→∞.

If then

 supt≥xg(t)∼g(x) and infa≤t≤xg(t)∼g(x),x→∞.
###### Definition 4

The Hermite polynomials , are given by

 Hm(x):=(−1)mexp(x22)dmdxmexp(−x22).

The first few Hermite polynomials are

 H0(x)=1, H1(x)=x, H2(x)=x2−1, H3(x)=x3−3x.

The Hermite polynomials , form a complete orthogonal system in the Hilbert space , where

is the probability density function of the standard normal distribution.

An arbitrary function possesses the mean-square convergent expansion

 S(ω)=∞∑j=0CjHj(ω)j!,Cj:=∫RS(ω)Hj(ω)ϕ(ω)dω. (1)

By Parseval’s identity

 ∞∑j=0C2jj!=∫RS2(ω)ϕ(ω)dω.
###### Definition 5

taqqu1975weak Let and there exists an integer , such that for all , but . Then is called the Hermite rank of and is denoted by

It is assumed that all random variables are defined on a fixed probability space . We consider a measurable mean-square continuous zero-mean homogeneous isotropic real-valued random field , with the covariance function

 B(r):=E(ξ(0)ξ(x)),x∈Rn,r=∥x∥.

It is well known that there exists a bounded non-decreasing function , , (see ivanov1989statistical , yadrenko1983spectral ) such that

 B(r)=∫∞0Yn(ru)dΦ(u),

where the function is defined by

 Yn(u):=2(n−2)/2Γ(n2)J(n−2)/2(u)u(2−n)/2,u⩾0,

where is the Bessel function of the first kind of order , see leonenko1999limit , yadrenko1983spectral . The function is called the isotropic spectral measure of the random field .

###### Definition 6

If there exists a function , such that

 un−1f(u)∈L1([0,∞)),Φ(u)=2πn/2/Γ(n/2)∫u0zn−1f(z)dz,

then the function is called the isotropic spectral density of the field .

The field with an absolutely continuous spectrum has the following isonormal spectral representation

 ξ(x)=∫Rnei⟨λ,x⟩√f(∥λ∥)W(dλ), (2)

where

is the complex Gaussian white noise random measure on

, see ivanov1989statistical , leonenko1999limit , yadrenko1983spectral .

Note, that by (2.1.8) leonenko1999limit we get and

 E(Hm1(ξ(x))Hm2(ξ(y)))=δm2m1m1!Bm1(∥x−y∥),x,y∈Rn,

where is the Kronecker delta function.

###### Definition 7

A random process , is called self-similar with parameter , if for any it holds .

If , is a self-similar process with parameter such that and , then , 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 , be a homogeneous isotropic Gaussian random field with and the covariance function , such that and

 B(x)=E(ξ(0)ξ(x))=∥x∥−αL0(∥x∥),α>0,

where is a function slowly varying at infinity.

If , then the covariance function satisfying Assumption 1 is not integrable, which corresponds to the long-range dependence caseanh2015rate .

The notation will be used to denote a Jordan-measurable compact bounded set, such that , and contains the origin in its interior. Let , be the homothetic image of the set , with the centre of homothety at the origin and the coefficient , that is and .

Let and denote the random variables and by

 Kκ:=∫△(r)S(ξ(x))dxandKr,κ:=Cκκ!∫△(r)Hκ(ξ(x))dx,

where is given by (1).

###### Theorem 3.1

leonenko2014sojourn Suppose that , satisfies Assumption 1 and . If a limit distribution exists for at least one of the random variables

 Kr√VarKrandKr,κ√VarKr,κ,

then the limit distribution of the other random variable also exists, and the limit distributions coincide when .

By Theorem 3.1 it is enough to study to get asymptotic distributions of . Therefore, we restrict our attention only to .

###### Assumption 2

The random field , has the isotropic spectral density

 f(∥λ∥)=c1(n,α)∥λ∥α−nL(1∥λ∥),

where and 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

will be used to denote the Fourier transform of the indicator function of the set

, i.e.

 KΔ(x):=∫Δei⟨u,x⟩du,x∈Rn. (3)
###### Theorem 3.2

leonenko2014sojourn Let , be a homogeneous isotropic Gaussian random field. If Assumptions 1 and 2 hold, then for the random variables

 Xr,κ(Δ):=rκα/2−nL−κ/2(r)∫Δ(r)Hκ(ξ(x))dx

converge weakly to

 Xκ(Δ):=cκ/21(n,α)∫′RnκKΔ(λ1+⋯+λκ)W(dλ1)⋯W(dλκ)∥λ1∥(n−α)/2⋯∥λκ∥(n−α)/2.

Here

denotes the multiple Wiener-Itô integral with respect to a Gaussian white noise measure, where the diagonal hyperplanes

, are excluded from the domain of integration.

###### Assumption 3

ivanov1989statistical Let be a radial continuous function positive for and such that there exists

 ~C:=κ!limr→∞∫Δ∫Δϑ(r∥x∥)ϑ(r∥y∥)dxdyϑ2(r)∥x−y∥ακ∈(0,∞),

where .

Let , where is from Assumption 2. In leonenko1999limit and Section 10.2 ivanov1989statistical the case when the function is continuous in a neighborhood of zero, bounded on and , was studied. It was assumed that there is a function such that

 ∫Rnκκ∏j=1∥λj∥α−n∣∣∣∫Δ(t1/n)ei⟨λ1+⋯+λκ,x⟩¯ϑ(x)dx∣∣∣2κ∏j=1dλj<∞

and

 limr→∞∫Rnκ∣∣∣∫Δ(t1/n)ei⟨λ1+⋯+λκ,x⟩(ϑ(r∥x∥)ϑ(r)κ∏j=1√u(∥λj∥r−1)u(0)−¯ϑ(x))dx∣∣∣2 ×κ∏j=1∥λj∥α−nκ∏j=1dλj=0,

for all .

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

 Yr,κ(t):=(rn−κα/2ϑ(r)√~Ccκ1(n,α)uκ(0))−1∫Δ(rt1/n)ϑ(∥x∥)Hκ(ξ(x))dx (4)

converge weakly to finite-dimensional distributions of the processes

 Yκ(t):=1√~Ccκ1(n,α)∫′RnκKΔ(t1/n)(λ1+⋯+λκ;¯ϑ)∏κj=1W(dλj)∏κj=1∥λj∥(n−α)/2,

as , where and

 KΔ(t1/n)(λ;¯ϑ):=∫Δ(t1/n)ei⟨λ,x⟩¯ϑ(x)dx.

## 4 Limit theorems for functionals of filtered fields

This section derives the generalisation of Theorem 3.3 when the integrand in (4) is replaced by a filtered random field.

###### Assumption 4

Let be a measurable real-valued homogeneous function of degree  and be a bounded uniformly continuous function such that in some neighberhood of zero and .

We define the filtered random field , as

 V(x)=∫RnG(∥y−x∥)Hκ(ξ(y))dy=∫RnG(∥y∥)Hκ(ξ(x+y))dy, (5)

where

 G(∥x∥):=1(2π)n∫Rne−i⟨x,u⟩h(∥u∥)g(∥u∥)du (6)

is the Fourier transform of .

###### Remark 1

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

 Hκ(ξ(x+y))=∫′Rnκei⟨λ1+⋯+λκ,x+y⟩κ∏j=1√f(∥λj∥)κ∏j=1W(dλj) (7)

it follows that

 V(x)=∫RnG(∥y∥)∫′Rnκei⟨λ1+⋯+λκ,x+y⟩κ∏j=1√f(∥λj∥)κ∏j=1W(dλj)dy =∫Rnei⟨λ1+⋯+λκ,x⟩∫′Rnκei⟨λ1+⋯+λκ,y⟩G(∥y∥)dyκ∏j=1(√f(∥λj∥)W(dλj)) =∫′Rnκei⟨λ1+⋯+λκ,x⟩^G(λ1+⋯+λκ)κ∏j=1√f(∥λj∥)κ∏j=1W(dλj),

where is the Fourier transform of the function 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 is

 V(x) =∫′Rnκei⟨λ1+⋯+λκ,x⟩h(∥λ1+⋯+λκ∥)g(∥λ1+⋯+λκ∥) ×κ∏j=1√f(∥λj∥)κ∏j=1W(dλj)=h(1)∫′Rnκei⟨λ1+⋯+λκ,x⟩∥λ1+⋯+λκ∥β ×g(∥λ1+⋯+λκ∥)κ∏j=1√f(∥λj∥)κ∏j=1W(dλj).

Therefore, it follows that the covariance of is

 Cov(V(x),V(y)) =h2(1)∫Rnκei⟨λ1+⋯+λκ,x−y⟩∥λ1+⋯+λκ∥2β ×g2(∥λ1+⋯+λκ∥)κ∏j=1f(∥λj∥)dλj. (8)
###### Remark 2

By the homogeneity of and Lemma 3 in leonenko2014sojourn it holds

 I1(α):=∫Rn|KΔ(λ)|2h2(∥λ∥)dλ∥λ∥n−α=h2(1)∫Rn|KΔ(λ)|2dλ∥λ∥n−α−2β<∞,

for and .

###### Lemma 1

If , are positive constants such that it holds
and , then

 Iκ(τ1,…,τκ): =∫Rnκ∣∣KΔ(λ1+⋯+λκ)∣∣2∥λ1+⋯+λκ∥2β ×∏κj=1dλj∥λ1∥n−τ1⋯∥λκ∥n−τκ<∞.
###### Proof

For we have and by Remark 2 we get the statement of the Lemma.

For , let us use the change of variables , where

. Then, by applying the recursive estimation routine we get

 Iκ(τ1,…,τκ)=∫Rn(κ−1)|KΔ(λ1+⋯+λκ−2+u)|2 ×∫Rn∥λ1+⋯+λκ−2+u∥2βdλκ−1∥λκ−1∥n−τκ−1∥u−λκ−1∥n−τκdλ1⋯dλκ−2du∥λ1∥n−τ1⋯∥λκ−2∥n−τκ−2
 =∫Rn(∫Rn(κ−2)|KΔ(λ1+⋯+λκ−2+u)|2∥λ1+⋯+λκ−2+u∥2β∏κ−2j=1dλj∥λ1∥n−τ1⋯∥λκ−2∥n−τκ−2∥u∥n−τκ−1−τκ ×∫Rnd~λκ−1∥~λκ−1∥n−τκ−1∥∥u∥u∥−~λκ−1∥∥n−τκ)du. (9)

Note, that the second integrand in (4) is unbounded at and (in this case ). If we split into the regions , , and we get

 ∫Rnd~λκ−1∥~λκ−1∥n−τκ−1∥∥u∥u∥−~λκ−1∥∥n−τκ≤sup~λκ−1∈A1∥∥u∥u∥−~λκ−1∥∥τκ−n +∫A3∥∥~λκ−1∥∥τκ−1−n|∥∥~λκ−1∥∥−1|τκ−nd~λκ−1≤(12)τκ−n∫1/20ρτκ−1−1dρ +(12)τκ−1−n∫A2−u∥u∥∥^λκ−1∥τκ−nd^λκ−1+∫∞3/2ρτκ−1−1(ρ−1)τκ−ndρ ≤C+(12)τκ−1−n∫5/20ρτκ−1dρ+∫∞1/2d^ρ^ρn+1−τκ−τκ−1=C<∞,

where , is a -dimensional ball with center and radius .

Hence, by (4) and Remark 2

 Iκ(τ1,…,τκ) ≤CIκ−1(τ1,…,τκ−2,τκ−1+τκ) ≤⋯≤CI1(κ∑i=1τi)≤C∫Rn|KΔ(u)|2du∥u∥n−∑κi=1τi−2β<∞, (10)

which completes the proof.∎

###### Lemma 2

The following integral is finite

 Jκ:=∫Rnκ∣∣^G(λ1+⋯+λκ)∣∣2κ∏i=1f(∥λi∥)dλi<∞.
###### Proof

As is an isotropic spectral density we can rewrite as

 Jκ =∫Rn(κ−1)∫Rn∣∣^G((λ1+⋯+λκ−1)+λκ)∣∣2f(∥−λκ∥)dλκκ−1∏i=1f(∥λi∥)dλi (11)

Note that