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 inrosenblatt1961independence 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 inveillette2013 . 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.
davydov1970invariance Let , where (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
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.
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
in a sense of convergence of finite-dimensional distributions.
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
satisfies the non-central limit theorem, that is for some normalising it holds
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 .
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
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.
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 ofto 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
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.
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.
A real-valued function is homogeneous of degree if for all .
bingham1987regular A measurable function is slowly varying at infinity if for all ,
By the representation theorem [bingham1987regular, , Theorem 1.3.1], there exists such that for all the function can be written in the form
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 .
bingham1987regular A measurable function is regularly varying at infinity, denoted , if there exists such that, for all , it holds that
[bingham1987regular, , Theorem 1.5.3] Let , and choose so that is locally bounded on . If then
The Hermite polynomials , are given by
The first few Hermite polynomials are
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
By Parseval’s identity
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
where the function is defined by
If there exists a function , such that
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
is the complex Gaussian white noise random measure on, see ivanov1989statistical , leonenko1999limit , yadrenko1983spectral .
Note, that by (2.1.8) leonenko1999limit we get and
where is the Kronecker delta function.
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.
Let , be a homogeneous isotropic Gaussian random field with and the covariance function , such that and
where is a function slowly varying at infinity.
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
where is given by (1).
By Theorem 3.1 it is enough to study to get asymptotic distributions of . Therefore, we restrict our attention only to .
The random field , has the isotropic spectral density
where and is a locally bounded function which is slowly varying at infinity.
will be used to denote the Fourier transform of the indicator function of the set, i.e.
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.
ivanov1989statistical Let be a radial continuous function positive for and such that there exists
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
for all .
Under these assumptions the following result was obtained.
4 Limit theorems for functionals of filtered fields
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
is the Fourier transform of .
Note, that from the isonormal spectral representation (2) and the Itô formula
it follows that
By the homogeneity of and Lemma 3 in leonenko2014sojourn it holds
for and .
If , are positive constants such that it holds
and , then
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
Note, that the second integrand in (4) is unbounded at and (in this case ). If we split into the regions , , and we get
where , is a -dimensional ball with center and radius .
The following integral is finite
As is an isotropic spectral density we can rewrite as