Limit theorems for filtered long-range dependent random fields

manuscript No. (will be inserted by the editor)

Limit theorems for filtered long-range dependent random fields

arXiv:1812.07290v1 [math.PR] 18 Dec 2018

T. Alodat · N. Leonenko · A. Olenko

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 ∈ ( 12 , 1). In this work we also obtain convergence for the case H ∈ (0, 12 ) and show how the Hurst parameter H can depend on the shape of the observation windows. Various examples are presented. Keywords filtered random fields · long-range dependence · self-similar processes · non-central limit theorem · Hurst parameter Mathematics Subject Classification (2010) 60G60 · 60F17 · 60F05 · 60G35 T. Alodat Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia E-mail: [email protected] N. Leonenko School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF2 4YH, UK E-mail: [email protected] A. Olenko Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia E-mail: [email protected]

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1 Introduction Over the last four decades, several studies dealt with various functionals of random fields and their asymptotic behaviour [4–6, 10, 11, 15, 19, 21, 36, 38]. 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 [2, 11, 22, 24, 28–30] 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 [33] 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 [10, 35–38]. This line of studies attracted much attention, for example, in [31] 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 [40]. The L´evy-Khintchine formula and asymptotic properties of the L´evy measure, were also addressed in [22]. Some weighted functionals for long-range dependent random fields were considered and limit theorems were investigated in a number of papers, including [14, 16, 29]. Linear stochastic processes and random fields obtained as outputs of filters are popular models in various applications, see [3, 17, 18, 41]. 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 [9] for discrete time linear stochastic processes. P Theorem 1 [9] Let V (t) = j∈Z Gt−j ξj , t ∈ Z, where ξj is a sequence of i.i.d random variables with zero mean and finite variance (the {ξj } are not necessarily Gaussian). Suppose that Gj is a real-valued sequence satisfying P P (d) (d) 2 := rt=1 V (t). If V arXr = r2H L2 (r) as r → ∞, j∈Z Gj < ∞ and let Xr where H ∈ (0, 1) and the function L(·) is a slowly varying at infinity, then Xr(d) (t)

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

in the sense of convergence of finite-dimensional distributions, where BH (t), t > 0, is the fractional Brownian motion with
zero mean and the covariance function BH (t, s) = 12 |t|2H + |s|2H − |t − s|2H , t, s > 0, 0 < H < 1.

Limit theorems for filtered long-range dependent random fields

3

One can obtain an analogous result for the case of continuous time. R Theorem 2 Let V (t) = R G(t − s)ξ(s)ds, t ∈ R, be a linear filtered process, where ξ(t), t ∈ R, be a mean-square continuous stationary in the wide sense process with zero mean and finite variance. Suppose that G(t), t ∈ R, is a Rr R (c) non-random function, such that R G2 (t)dt < ∞. Let Xr := 0 V (s)ds. If (c)

V arXr = r2H L2 (r), as r → ∞, where H ∈ (0, 1) and L(·) is slowly varying at infinity, then Z rt 1 D (c) Xr (t) = H V (s)ds → BH (t), t > 0, as r → ∞, r L(r) 0 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 [24] and Alodat and Olenko [2]. It was Rossenblatt [34] (see also Major [27], Taqqu [11]) who first proved that for a discrete-time Gaussian stochastic process {ξj , j ∈ Z}, with zero mean and long-range dependence and the κ-th Hermite polynomials Hκ , the non-linear filtered process X Vκ (t) = Gt−j Hκ (ξj ), j∈Z

satisfies the non-central limit theorem, that is for some normalising Ar it holds [rt] 1 X D Vκ (s) → Yκ (t), t > 0, as r → ∞, Ar s=1

where Yκ (t) is a self-similar process with the Hurst parameter H ∈ (0, 1) (non-Gaussian, if κ ≥ 2). The limit processes Yκ (t), t > 0, are given in terms of κ-fold Wiener-Itˆ o stochastic integrals, and are the fractional Brownian motions with the Hurst parameter H ∈ (0, 1) if κ = 1. The aim of this paper is to give an extension of the results of Rossenblatt [34], Major [27], Taqqu [11] 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. [1,23,25,26]), we study the asymptotic behaviour of integrals of the form Z −1 V (x)dx, t ∈ [0, 1], as r → ∞, dr ∆(rt1/n )

where V (x), x ∈ Rn , is a random field, ∆ ⊂ Rn is an observation window and dr is a normalising factor. The case when the limit process is self-similar with parameter H ∈ (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

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be classified according to the range of H to the Brownian motion (H = 0.5), a short-memory anti-persistent stochastic process (H ∈ (0, 0.5)) and a longmemory stochastic process (H ∈ (0.5, 1)). These three cases correspond to the three types of behaviour called f1 noise, ultraviolet and infrared catastrophes by Mandelbrot and Taqqu [39]. 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 ∈ (0.5, 1), see [14, 20, 29, 30, 36, 38] and references therein. However, there are only few results where asymptotic processes have H ∈ (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, see [12, 13]. The example of a non-Gaussian self-similar process with H ∈ (0, 0.5) was given by Rossenblatt [34] where the asymptotic of quadratic functions of a long-range Gaussian stationary sequence was investigated. The result was generalised in [27] 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 Z G(ky − xk)S(ξ(y))dy, V (x) := Rn

where G(·), S(·) are non-random functions and ξ(·) is a long-range dependent random field. In the limit theorems obtained in this paper the asymptotic processes have the self-similar parameters H ∈ (γ(∆), 1), where γ(∆) ≥ 0 depends on the geometry of the set ∆ ⊂ Rn . In the one-dimensional case d = 1, γ(∆) = 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 | · | and k · k are used for the Lebesque measure and the Euclidean distance in Rn , n ≥ 1, respectively. The symbols C, ǫ and δ (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, ∞) → R is homogeneous of degree β if h(ax) = aβ h(x) for all a, x > 0.

Limit theorems for filtered long-range dependent random fields

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Definition 2 [7] A measurable function L : (0, ∞) → (0, ∞) is slowly varying at infinity if for all t > 0, L(tr) = 1. lim r→∞ L(r) By the representation theorem [7, Theorem 1.3.1], there exists C > 0 such that for all r ≥ C the function L(·) can be written in the form   Z r ζ2 (u) L(r) = exp ζ1 (r) + du , u C where ζ1 (·) and ζ2 (·) are such measurable and bounded functions that ζ2 (r) → 0 and ζ1 (r) → C0 , (C0 < ∞), when r → ∞. If L(·) varies slowly, then ra L(r) → ∞, and r−a L(r) → 0 for an arbitrary a > 0 when r → ∞, see Proposition 1.3.6 [7]. Definition 3 [7] A measurable function g : (0, ∞) → (0, ∞) is regularly varying at infinity, denoted g(·) ∈ Rτ , if there exists τ such that, for all t > 0, it holds that lim

r→∞

g(tr) = tτ . g(r)

Theorem 3 [7, Theorem 1.5.3] Let g(·) ∈ Rτ , and choose a ≥ 0 so that g is locally bounded on [a, ∞). If τ > 0 then sup g(t) ∼ g(x) and inf g(t) ∼ g(x),

a≤t≤x

t≥x

x → ∞.

If τ < 0 then sup g(t) ∼ g(x) and t≥x

inf g(t) ∼ g(x),

a≤t≤x

x → ∞.

Definition 4 The Hermite polynomials Hm (x), m ≥ 0, are given by  2  2 m d x x . exp − Hm (x) := (−1)m exp 2 dxm 2 The first few Hermite polynomials are H0 (x) = 1, H1 (x) = x, H2 (x) = x2 − 1, H3 (x) = x3 − 3x. The Hermite polynomials Hm (x), m ≥ 0, form sys R a complete orthogonal tem in the Hilbert space L2 (R, φ(ω)dω) = S : R S 2 (ω)φ(ω)dω < ∞ , where φ(ω) is the probability density function of the standard normal distribution. An arbitrary function S(ω) ∈ L2 (R, φ(ω)dω) possesses the mean-square convergent expansion Z ∞ X Cj Hj (ω) S(ω)Hj (ω)φ(ω)dω. (1) , Cj := S(ω) = j! R j=0

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By Parseval’s identity ∞ X Cj2 j=0

j!

=

Z

S 2 (ω)φ(ω)dω.

R

Definition 5 [36] Let S(ω) ∈ L2 (R, φ(ω)dω) and there exists an integer κ > 1, such that Cj = 0 for all 0 < j ≤ κ − 1, but Cκ 6= 0. Then κ is called the Hermite rank of S(·) and is denoted by HrankS(·). It is assumed that all random variables are defined on a fixed probability space (Ω, F, P). We consider a measurable mean-square continuous zero-mean homogeneous isotropic real-valued random field ξ (x) , x ∈ Rn , with the covariance function B (r) := E (ξ(0)ξ(x)) ,

x ∈ Rn ,

r = kxk.

It is well known that there exists a bounded non-decreasing function Φ (u), u ≥ 0, (see [14, 42]) such that Z ∞ B (r) = Yn (ru) dΦ (u) , 0

where the function Yn (·) , n ≥ 1, is defined by n Yn (u) := 2(n−2)/2 Γ J(n−2)/2 (u)u(2−n)/2 , u > 0, 2 where J(n−2)/2 (·) is the Bessel function of the first kind of order (n − 2)/2, see [20, 42]. The function Φ (·) is called the isotropic spectral measure of the random field ξ (x) , x ∈ Rn . Definition 6 If there exists a function f (u), u ∈ [0, ∞), such that Z u un−1 f (u) ∈ L1 ([0, ∞)), Φ(u) = 2π n/2 /Γ (n/2) z n−1 f (z)dz, 0

then the function f (·) is called the isotropic spectral density of the field ξ (x). The field ξ (x) with an absolutely continuous spectrum has the following isonormal spectral representation Z p (2) eihλ,xi f (kλk)W (dλ), ξ (x) = Rn

where W (·) is the complex Gaussian white noise random measure on Rn , see [14, 20, 42]. Note, that by (2.1.8) [20] we get E (Hm (ξ(x))) = 0 and m2 m1 !B m1 (kx − yk), E (Hm1 (ξ(x))Hm2 (ξ(y))) = δm 1

m2 is the Kronecker delta function. where δm 1

x, y ∈ Rn ,

Definition 7 A random process X(t), t > 0, is called self-similar with paD rameter H > 0, if for any a > 0 it holds X(at) = aH X(t). If X(t), t > 0, is a self-similar process with parameter H > 0 such that E(X(t)) = 0 and E(X 2 (t)) < ∞, then B(at, as) = a2H B(t, s), see [20].

Limit theorems for filtered long-range dependent random fields

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3 Assumptions and auxiliary results This section introduces assumptions and results from the spectral and correlation theory of random fields. Assumption 1 Let ξ(x), x ∈ Rn , be a homogeneous isotropic Gaussian random field with Eξ(x) = 0 and the covariance function B(x), such that B(0) = 1 and B(x) = E (ξ (0) ξ (x)) = kxk−α L0 (kxk) , α > 0, where L0 (k · k) is a function slowly varying at infinity.

If α ∈ (0, n), then the covariance function B(x) satisfying Assumption 1 is not integrable, which corresponds to the long-range dependence case [4]. The notation ∆ ⊂ Rn will be used to denote a Jordan-measurable compact bounded set, such that |∆| > 0, and ∆ contains the origin in its interior. Let ∆(r), r > 0, be the homothetic image of the set ∆, with the centre of homothety at the origin and the coefficient r > 0, that is |∆(r)| = rn |∆| and ∆ = ∆(1). Let S(ω) ∈ L2 (R, φ(ω)dω) and denote the random variables Kκ and Kr,κ by Z Z Cκ Hκ (ξ (x)) dx, Kκ := S (ξ (x)) dx and Kr,κ := κ! △(r) △(r)

where Cκ is given by (1).

Theorem 4 [21] Suppose that ξ (x) , x ∈ Rn , satisfies Assumption 1 and HrankS(·) = κ ≥ 1. If a limit distribution exists for at least one of the random variables Kr,κ Kr √ and p , V arKr V arKr,κ then the limit distribution of the other random variable also exists, and the limit distributions coincide when r → ∞.

By Theorem 4 it is enough to study Kr,κ to get asymptotic distributions of Kκ . Therefore, we restrict our attention only to Kr,κ . Assumption 2 The random field ξ (x) , x ∈ Rn , has the isotropic spectral density   1 , f (kλk) = c1 (n, α) kλkα−n L kλk
α n/2
where α ∈ (0, n), c1 (n, α) := Γ n−α /2 π Γ α2 , and L(k · k) ∼ L0 (k · k) 2 is a locally bounded function which is slowly varying at infinity.

One can find more details on relations between Assumptions 1 and 2 in [4, 5]. The function K∆ (x) will be used to denote the Fourier transform of the indicator function of the set ∆, i.e. Z eihu,xi du, x ∈ Rn . (3) K∆ (x) := ∆

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Theorem 5 [21] Let ξ (x) , x ∈ Rn , be a homogeneous isotropic Gaussian random field. If Assumptions 1 and 2 hold, then for r → ∞ the random variables Z Hκ (ξ(x)) dx Xr,κ (∆) := rκα/2−n L−κ/2 (r) ∆(r)

converge weakly to Xκ (∆) :=

κ/2 c1 (n, α)

Z

′

Rnκ

K∆ (λ1 + · · · + λκ )

W (dλ1 ) · · · W (dλκ ) . kλ1 k(n−α)/2 · · · kλκ k(n−α)/2

R′ Here Rnκ denotes the multiple Wiener-Itˆ o integral with respect to a Gaussian white noise measure, where the diagonal hyperplanes λi = ±λj , i, j = 1, . . . , κ, i 6= j, are excluded from the domain of integration. Assumption 3 [14] Let ϑ(x) = ϑ(kxk) be a radial continuous function positive for kxk > 0 and such that there exists Z Z ϑ(rkxk)ϑ(rkyk)dxdy ∈ (0, ∞) , C˜ := κ! lim r→∞ ∆ ∆ ϑ2 (r)kx − ykακ where α ∈ (0, n/κ).  1 , where L(·) is from Assumption 2. In [20] Let u(kλk) := c1 (n, α) L kλk and Section 10.2 [14] the case when the function u(kλk) is continuous in a neighborhood of zero, bounded on (0, ∞) and u(0) 6= 0, was studied. It was ¯ assumed that there is a function ϑ(kxk) such that Z 2 κ Z κ Y Y ¯ kλj kα−n dλj < ∞ eihλ1 +···+λκ ,xi ϑ(x)dx 

Rnκ j=1

∆(t1/n )

j=1

and

lim

r→∞

×

Z

κ Y

j=1

R



κ ϑ(rkxk) Y eihλ1 +···+λκ ,xi  ϑ(r) j=1 ∆(t1/n )

Z nκ

kλj kα−n

κ Y

s

u(kλj kr u(0)

−1

)

dλj = 0,



2 ¯  dx − ϑ(x)

j=1

for all t ∈ [0, 1]. Under these assumptions the following result was obtained. Theorem 6 [14] If Assumption 3 holds, then the finite-dimensional distributions of the random processes −1 Z  q ˜ κ (n, α)uκ (0) ϑ(kxk)Hκ (ξ(x)) dx Yr,κ (t) := rn−κα/2 ϑ(r) Cc 1 ∆(rt1/n )

(4)

Limit theorems for filtered long-range dependent random fields

9

converge weakly to finite-dimensional distributions of the processes Qκ Z ′
1 j=1 W (dλj ) ¯ Q q , K∆(t1/n ) λ1 + · · · + λκ ; ϑ Yκ (t) := κ (n−α)/2 ˜ κ (n, α) Rnκ j=1 kλj k Cc 1

as r → ∞, where α ∈ 0, min

n n+1 κ, 2


K∆(t1/n ) λ; ϑ¯ :=





and

Z

¯ eihλ,xi ϑ(x)dx.

∆(t1/n )

4 Limit theorems for functionals of filtered fields This section derives the generalisation of Theorem 6 when the integrand ϑ(·)Hκ (·) in (4) is replaced by a filtered random field. Assumption 4 Let h : [0, ∞) → R be a measurable real-valued homogeneous function of degree β and g : [0, ∞) → R be a bounded uniformly continuous function such that g(0) 6= 0 in some neighberhood of zero and R 2 2 h (kuk)g (kuk)du < ∞. n R We define the filtered random field V (x), x ∈ Rn , as Z Z G(kyk)Hκ (ξ(x + y))dy, G(ky − xk)Hκ (ξ(y))dy = V (x) =

(5)

Rn

Rn

where G(kxk) :=

1 (2π)n

Z

e−ihx,ui h(kuk)g(kuk)du

(6)

Rn

is the Fourier transform of h(·)g(·). Remark 1 Note, that from the isonormal spectral representation (2) and the Itˆ o formula Z ′ κ κ q Y Y W (dλj ) (7) f (kλj k) Hκ (ξ(x + y)) = eihλ1 +···+λκ ,x+yi Rnκ

it follows that Z Z V (x) = G(kyk) Rn

=

Z

eihλ1 +···+λκ ,xi

Rn

=

Z

′

Rnκ

′

e

Rnκ

Z

′

Rnκ

j=1

j=1

ihλ1 +···+λκ ,x+yi

κ q κ Y Y W (dλj )dy f (kλj k)

j=1

j=1

 κ q Y eihλ1 +···+λκ ,yi G(kyk)dy f (kλj k)W (dλj ) j=1

κ q κ Y Y ˆ 1 + · · · + λκ ) eihλ1 +···+λκ ,xi G(λ f (kλj k) W (dλj ), j=1

j=1

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ˆ is the Fourier transform of the function G(·) that is defined by (6) where G(·) and the stochastic Fubini’s theorem [32, 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 Z ′ V (x) = eihλ1 +···+λκ ,xi h(kλ1 + · · · + λκ k)g(kλ1 + · · · + λκ k) ×

Rnκ κ Yq

j=1

f (kλj k)

κ Y

W (dλj ) = h(1)

Z

′

Rnκ

j=1

× g(kλ1 + · · · + λκ k)

eihλ1 +···+λκ ,xi kλ1 + · · · + λκ kβ

κ q κ Y Y W (dλj ). f (kλj k)

j=1

j=1

Therefore, it follows that the covariance of V (x) is Z eihλ1 +···+λκ ,x−yi kλ1 + · · · + λκ k2β Cov(V (x), V (y)) = h2 (1) Rnκ

2

× g (kλ1 + · · · + λκ k)

κ Y

j=1

f (kλj k)dλj .

(8)

Remark 2 By the homogeneity of h(·) and Lemma 3 in [21] it holds Z Z h2 (kλk) dλ dλ 2 |K∆ (λ) |2 I1 (α) := |K∆ (λ) |2 = h (1) < ∞, n−α n−α−2β kλk kλk n n R R for α ∈ (0, n − 2β) and β < n/2. Lemma 1 If τ1 , . . . , τκ , κ ≥ 1, are positive constants such that it holds Pκ τ < n − 2β and β < n/2, then i=1 i Z K∆ (λ1 + · · · + λκ ) 2 kλ1 + · · · + λκ k2β Iκ (τ1 , . . . , τκ ) : = Rnκ Qκ j=1 dλj < ∞. × kλ1 kn−τ1 · · · kλκ kn−τκ Proof For κ = 1 we have τ1 ∈ (0, n−2β) and by Remark 2 we get the statement of the Lemma. ˜ κ−1 = λκ−1 /kuk, where For κ > 1, let us use the change of variables λ u = λκ + λκ−1 . Then, by applying the recursive estimation routine we get Z |K∆ (λ1 + · · · + λκ−2 + u) |2 Iκ (τ1 , . . . , τκ ) = Rn(κ−1) Z kλ1 + · · · + λκ−2 + uk2β dλκ−1 dλ1 · · · dλκ−2 du × n−τ n−τ n−τ κ κ−1 ku − λκ−1 k kλ1 k 1 · · · kλκ−2 kn−τκ−2 Rn kλκ−1 k =

Z 

Rn

Z

Rn(κ−2)

|K∆ (λ1 + · · · + λκ−2 + u) |2 kλ1 + · · · + λκ−2 + uk2β kλ1

kn−τ1

· · · kλκ−2

kn−τκ−2 kukn−τκ−1 −τκ

Qκ−2 j=1

dλj

Limit theorems for filtered long-range dependent random fields

×

Z

Rn

11

 ˜ κ−1 dλ du.

˜ κ−1 n−τκ ˜ κ−1 kn−τκ−1 u − λ kλ kuk

(9)

˜ κ−1 k = 0 and Note, that the second integrand in (9) is unbounded at kλ ˜ κ−1 = u/kuk (in this case kλ ˜ κ−1 k = 1). If we split Rn into the regions λ ˜ κ−1 ∈ Rn : 1 ≤ kλ ˜ κ−1 k < 3 }, and ˜ κ−1 ∈ Rn : kλ ˜ κ−1 k < 1 }, A2 := {λ A1 := {λ 2 2 2 ˜ κ−1 ∈ Rn : kλ ˜ κ−1 k ≥ 3 } we get A3 := {λ 2 ˜ κ−1

u

dλ ˜ κ−1 τκ −n −λ

n−τκ ≤ sup

u ˜ ˜ κ−1 kn−τκ−1

˜ κ−1 ∈A1 kuk Rn k λ λ kuk − λκ−1 Z Z ˜ κ−1 dλ τκ−1 −n τκ−1 −n ˜ ˜ ˜ kλκ−1 k dλκ−1 + sup kλκ−1 k ×

n−τκ

u ˜

˜ κ−1 ∈A2 A2 A1 λ kuk − λκ−1  τκ −n Z 1/2 Z

τκ−1 −n

˜ κ−1 ≤ 1

˜ λκ−1 − 1|τκ −n dλ | ˜ λκ−1 + ρτκ−1 −1 dρ 2 A3 0  τκ−1 −n Z Z ∞ 1 τ −n ˆ κ−1 kτκ −n dλ ˆ κ−1 + ρτκ−1 −1 (ρ − 1) κ dρ + u kλ 2 A2 − 3/2 kuk  τκ−1 −n Z 5/2 Z ∞ 1 dˆ ρ τκ −1 ≤C+ = C < ∞, ρ dρ + n+1−τκ −τκ−1 2 ρ ˆ 1/2 0

Z

u u = {λ ∈ Rn : λ+ ∈ A2 } ⊂ vn kuk kuk ball with center 0 and radius r. Hence, by (9) and Remark 2

where A2 −

5 2


, vn (r) is a n-dimensional

Iκ (τ1 , . . . , τκ ) ≤ CIκ−1 (τ1 , . . . , τκ−2 , τκ−1 + τκ ) ! Z κ X |K∆ (u) |2 du P τi ≤ C < ∞, ≤ · · · ≤ CI1 n− κ i=1 τi −2β Rn kuk i=1 which completes the proof.

(10) ⊓ ⊔

Lemma 2 The following integral is finite Z κ Y G(λ ˆ 1 + · · · + λκ ) 2 f (kλi k)dλi < ∞. Jκ := Rnκ

i=1

Proof As f (·) is an isotropic spectral density we can rewrite Jκ as Jκ =

Z

Z

=

Z

Z

Rn(κ−1)

Rn(κ−1)

Rn

Rn

κ−1 Y G((λ ˆ 1 + · · · + λκ−1 ) + λκ ) 2 f (k − λκ k)dλκ f (kλi k)dλi i=1

κ−1   Y G ˆ 2 ∗ f (λ1 + · · · + λκ−1 ) f (k − λi k)dλi . i=1

(11)

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ˆ 2 (·) ∈ L1 (Rn ) and f (·) ∈ L1 (Rn ). Hence, by Young’s theorem [8] Note that |G|   ˆ 2 (·) = G ˆ 2 ∗ f (·) ∈ L1 (Rn ). Therefore, using convolutions it follows that G 1

as in (11) we obtain

κ−2 Y

Jκ =

Z

Z

ˆ 21 (λ1 + · · · + λκ−1 )f (k − λκ−1 k)dλκ−1 G

=

Z

Z

κ−2   Y ˆ 21 ∗ f (λ1 + · · · + λκ−2 ) f (k − λi k)dλi G

=

Z

Z

Rn(κ−2)

Rn(κ−2)

Rn(κ−2)

=

Z

Rn

Rn

Rn

Rn

i=1

f (k − λi k)dλi

i=1

ˆ 22 (λ1 + · · · + λκ−2 ) G

κ−2 Y i=1

f (k − λi k)dλi = · · · =

ˆ 2 (λ1 )f (k − λ1 k)dλ1 < ∞, G κ−1

ˆ 2 (·) := where G j+1 steps.



 ˆ 2 ∗ f (·) ∈ L1 (Rn ) by Young’s theorem and recursive G j ⊓ ⊔

Now we proceed to the main result. Theorem 7 Let ξ (x) , x ∈ Rn , be a random field satisfying Assumptions 1, 2 and functions g(·) and h(·) satisfy Assumption 4. Then, for r → +∞ the finite-dimensional distributions of Z rβ+κα/2−n L−κ/2 (r) Xr,κ (t) := V (x) dx, t ∈ [0, 1], κ/2 (2π)n c1 (n, α)g(0)h(1) ∆(rt1/n ) converge weakly to the finite-dimensional distributions of Z ′  kλ1 + · · · + λκ kβ Qκ W (dλj )  j=1 1/n Qκ , K∆ (λ1 + · · · + λκ ) t Xκ (t) := t (n−α)/2 kλ k nκ j R j=1   and β < n2 . where α ∈ 0, n−2β κ

Remark 3 By the representation (8) in Remark 1 of the covariance function of V (x) we obtain Cov (Xr,κ (t), Xr,κ (s)) = =

=

rβ+κα/2−n L−κ/2 (r) κ/2

!2 Z

(2π)n c1 (n, α)g(0)h(1) !2 Z rβ+κα/2−n L−κ/2 (r) κ/2

(2π)n c1 (n, α)g(0)

∆(rt1/n )

∆(rt1/n )

Z

Z

Cov (V (x), V (y)) dxdy

∆(rs1/n )

∆(rs1/n )

× kλ1 + · · · + λκ k2β g 2 (kλ1 + · · · + λκ k)

κ Y

j=1

Z

eihλ1 +···+λκ ,x−yi

Rnκ

f (kλj k)dλj dxdy

Limit theorems for filtered long-range dependent random fields

=

rβ+κα/2−n L−κ/2 (r) κ/2

(2π)n c1 (n, α)g(0)

!2 Z

Rnκ

Z

∆(rt1/n )

× kλ1 + · · · + λκ k2β g 2 (kλ1 + · · · + λκ k) =

rβ+κα/2 L−κ/2 (r) κ/2

(2π)n c1 (n, α)g(0)

!2

ts

Z

Rnκ

Z

κ Y

13

eihλ1 +···+λκ ,x−yi dxdy

∆(rs1/n )

j=1

f (kλj k)dλj

κ Y  f (kλj k) K∆ (λ1 + · · · + λκ ) rt1/n j=1

κ Y
dλj . × K∆ (λ1 + · · · + λκ ) rs1/n kλ1 + · · · + λκ k2β g 2 (kλ1 + · · · + λκ k) j=1

In particular, the variance of Xr,κ (t) is equal !2 Z rβ+κα/2 L−κ/2 (r) V ar (Xr,κ (t)) = t2 kλ1 + · · · + λκ k2β κ/2 Rnκ (2π)n c1 (n, α)g(0) κ  2  Y 1/n 2 f (kλj k)dλj . g (kλ + · · · + λ k) × K∆ (λ1 + · · · + λκ ) rt 1 κ j=1

Similarly, we get

Cov (Xκ (t), Xκ (s)) = ts

Z

Rnκ

  K∆ (λ1 + · · · + λκ ) t1/n

κ Y
× K∆ (λ1 + · · · + λκ ) s1/n kλ1 + · · · + λκ k2β kλj kα−n dλj j=1

and

V ar (Xκ (t)) = t

2

  kλ1 + · · · + λκ k2β Qκ dλj j=1 1/n K∆ (λ1 + · · · + λκ ) t 2 Qκ . n−α kλ k j Rnκ j=1

Z

Remark 4 Note, that for a > 0 we have Z ′   kλ1 + · · · + λκ kβ Qκ W (dλj ) j=1 1/n . Xκ (at) = at K∆ (λ1 + · · · + λκ ) (at) Qκ n−α 2 kλ k Rnκ j j=1

˜ j = a n1 λj , j = 1, . . . , κ, and the self-similarity of Using the transformation λ the Gaussian white noise we get β Z ′    kλ ˜1 + · · · + λ ˜ κ kβ a1− n 1/n ˜ ˜ λ + · · · + λ t K Xκ (at) =  t Qκ 1 κ ∆  κ(n−α)/2 (n−α)/2 ˜ 1 Rnκ j=1 kλj k a− n ×

κ Y

β 1 ˜ j ) = a1− κα 2n − n X (t). W (a− n dλ κ

j=1

Thus, the random process Xκ (t) is a self-similar with the Hurst parameβ ter H = 1 − κα 2n − n .

14

T. Alodat et al.

Proof By (5) the process Xr,κ (t) admits the following representation Xr,κ (t) =

rβ+κα/2−n L−κ/2 (r) κ/2

(2π)n c1 (n, α)g(0)h(1)

Z

∆(rt1/n )

Z

Rn

 G(kyk)Hκ (ξ(x + y))dy dx.

By (7) we obtain rβ+κα/2−n L−κ/2 (r)

Z

Z

G(kyk) κ/2 Rn (2π)n c1 (n, α)g(0)h(1) ∆(rt1/n ) Z ′   κ q κ Y Y ihλ1 +···+λκ ,x+yi × e W (dλj ) dy dx. f (kλj k)

Xr,κ (t) =

Rnκ

j=1

(12)

j=1

p Q By Assumption 2 it follows κj=1 f (kλj k) ∈ L2 (Rnκ ). By Assumption 4, (6) and Parseval’s theorem G(·) ∈ L2 (Rn ). So, one can apply the stochastic Fubini’s theorem to interchange the inner integrals in (12), see Theorem 5.13.1 in [32], which results in Xr,κ (t) =

rβ+κα/2−n L−κ/2 (r) κ/2

Z

c1 (n, α)g(0)h(1) ∆(rt1/n ) κ κ q Y Y W (dλj )dx. f (kλj k) ×

Z

′

Rnκ

ˆ 1 + · · · + λκ ) eihλ1 +···+λκ ,xi G(λ (13)

j=1

j=1

Note, that by Lemma 2 the integrand in (13) belongs to L2 (Rnκ ). Then, it follows from the stochastic Fubini’s theorem, and Assumption 4 that Xr,κ (t) =

rβ+κα/2−n L−κ/2 (r)

Z

κ/2

c1 (n, α)g(0)

′

Rnκ

K∆(rt1/n ) (λ1 + · · · + λκ )

× kλ1 + · · · + λκ kβ g(kλ1 + · · · + λκ k)

κ q κ Y Y W (dλj ), f (kλj k)

j=1

j=1

where K∆(rt1/n ) (λ) =

Z

eihλ,xi dx.

∆(rt1/n )


Note, that K∆(rt1/n ) (λ) = trn K∆ λrt1/n , where K∆ (·) is given by (3). Therefore, Xr,κ (t) = t

rβ+κα/2 L−κ/2 (r) κ/2

c1 (n, α)g(0)

Z

′

Rnκ

  K∆ (λ1 + · · · + λκ ) rt1/n

× kλ1 + · · · + λκ kβ g(kλ1 + · · · + λκ k)

κ κ q Y Y f (kλj k) W (dλj ).

j=1

j=1

Limit theorems for filtered long-range dependent random fields

15

Using the transformation λ(j) = rλj , j = 1, . . . , κ, and the self-similarity of the Gaussian white noise we get Xr,κ (t) = t

rβ+κα/2 L−κ/2 (r) κ/2

′

Z

K∆



  λ(1) + · · · + λ(κ) t1/n

Rnκ c1 (n, α)g(0) κ q 

β Y f (kλ(j) k/r) × r−1 λ(1) + · · · + λ(κ) j=1

κ 

 Y rκα/2 L−κ/2 (r)r−nκ/2 W (dλ(j) /r) = t × g r−1 λ(1) + · · · + λ(κ) κ/2 c1 (n, α)g(0) j=1 Z ′   

β × K∆ λ(1) + · · · + λ(κ) t1/n λ(1) + · · · + λ(κ) Rnκ

κ q κ  Y

 Y × f (kλ(j) k/r)g r−1 λ(1) + · · · + λ(κ) W (dλ(j) ) j=1

=t

r

j=1

κ(α−n)/2

−κ/2

L g(0)

(r)

Z

′

K∆

Rnκ



  λ(1) + · · · + λ(κ) t1/n

κ q

β Y × λ(1) + · · · + λ(κ) (kλ(j) k/r)α−n L(r/kλ(j) k) j=1

κ 

 Y W (dλ(j) ) × g r−1 λ(1) + · · · + λ(κ) j=1

′





K∆ λ + · · · + λ(κ) t1/n t

λ(1) + · · · + λ(κ) β Q = κ (j) (n−α)/2 g(0) Rnκ j=1 kλ k κ κ q  Y

 Y W (dλ(j) ). L(r/kλ(j) k)/L(r)g r−1 λ(1) + · · · + λ(κ) × Z

(1)

j=1

j=1

By the isometry property of multiple stochastic integrals

λ(1) + · · · + λ(κ) t1/n |2 Qκ Rr : = E (Xr,κ (t) − Xκ (t)) = t (j) n−α Rnκ j=1 kλ k 2

2β  Qr (λ(1) , . . . , λ(κ) ) − 1 dλ(1) · · · dλ(κ) , × λ(1) + · · · + λ(κ) 2

2

Z

|K∆

where


v κ   −1 (1) (κ) u uY g r λ + ···+ λ t L(r/kλ(j) k)/L(r). Qr λ(1) , . . . , λ(κ) := g(0) j=1 Note, that by Assumptions 2, 4, and properties of slowly varying functions Qr (λ(1) , . . . , λ(κ) ) pointwise converges to 1, when r → ∞.

16

T. Alodat et al.

Let us split Rnκ into the regions   Bµ := λ(1) , . . . , λ(κ) ∈ Rnκ : kλ(j) k ≤ 1, if µj = −1, (j)

and kλ

 k > 1, if µj = 1, j = 1, . . . , κ ,

where µ = (µ1 , . . . , µκ ) ∈ {−1, 1}κ is a binary vector of length κ. Then we can represent the integral Rr as

Z |K∆ λ(1) + · · · + λ(κ) t1/n |2 2 Qκ Rr = t (j) n−α ∪µ∈{−1,1}κ Bµ j=1 kλ k  2

2β   (1) Qr λ , . . . , λ(κ) − 1 dλ(1) · · · dλ(κ) . × λ(1) + · · · + λ(κ)


If λ(1) , . . . , λ(κ) ∈ Bµ we estimate the integrand as follows

2

 |K∆ λ(1) + · · · + λ(κ) t1/n |2

λ(1) + · · · + λ(κ) 2β Qr (λ(1) , . . . , λ(κ) ) − 1 Qκ (j) n−α j=1 kλ k



 2|K∆ λ(1) + · · · + λ(κ) t1/n |2

λ(1) +· · ·+λ(κ) 2β Q2r (λ(1) , . . . , λ(κ) ) + 1 Qκ ≤ (j) n−α j=1 kλ k



2|K∆ λ(1) + · · · + λ(κ) t1/n |2

λ(1) + · · · + λ(κ) 2β Qκ = (j) n−α j=1 kλ k  


κ r r µ δ κ g 2 r−1 λ(1) + · · · + λ(κ) Y (j) µj δ Y ( kλ(j) k ) j L( kλ(j) k ) , × 1 + kλ k g 2 (0) rµj δ L(r) j=1 j=1 where δ is an arbitrary positive number. Using the boundedness of the function g(·), we can write

2

 |K∆ λ(1) + · · · + λ(κ) t1/n |2

λ(1) + · · · + λ(κ) 2β Qr (λ(1) , . . . , λ(κ) ) − 1 Qκ (j) n−α j=1 kλ k ≤ 

2 K∆



2 λ(1) + · · · + λ(κ) t1/n

λ(1) + · · · + λ(κ) 2β Qκ (j) n−α j=1 kλ k

 κ (j) µj δ (j) Y (r/kλ k) L(r/kλ k) . kλ(j) kµj δ × 1 + C sup rµj δ L(r) (λ1 ,...,λκ )∈Bµ j=1 j=1 κ Y

By Theorem 3


−δ
supkλ(j) k≤1 r/kλ(j) k L r/kλ(j) k lim = 1; r→∞ r−δ L(r)

Limit theorems for filtered long-range dependent random fields

17

and
δ
supkλ(j) k>1 r/kλ(j) k L r/kλ(j) k = 1. lim r→∞ rδ L(r)
Therefore, there exists r0 > 0 such that for all r ≥ r0 and λ(1) , . . . , λ(κ) ∈ Bµ K ∆

(1)

(κ)




λ + ···+ λ t Qκ (j) n−α j=1 kλ k 2 K∆

(1)

1/n


2 2



λ(1) + · · · + λ(κ) 2β Qr (λ(1) , . . . , λ(κ) ) − 1 (κ)




1/n


2

λ(1) + · · · + λ(κ) 2β

λ + ··· + λ t Qκ (j) n−α j=1 kλ k



2C|K∆ λ(1) + · · · + λ(κ) t1/n |2

λ(1) + · · · + λ(κ) 2β . Q + (14) κ (j) n−α−µj δ j=1 kλ k   By Lemma 1, if we choose δ ∈ 0, min{α, n−2β − α} , the upper bound in κ ≤

(14) is an integrable function on each Bµ and hence on Rnκ too. By Lebesque’s dominated convergence theorem Rr → 0 as r → ∞, which completes the proof. ⊓ ⊔ 5 Examples

The objective of this section is to investigate the Hurst parameter H of the limit process Xκ (t) in Theorem 7. The section provides simple examples where the range (γ(∆), 1) for H is explicitly specified depending on the observation window ∆ ⊂ Rn . β Recall that H = 1 − κα 2n − n and I1 (κα) is defined as I1 (κα) = C

Z

Rn

|K∆ (λ) |2 dλ . kλkn−κα−2β

Example 1 Let n = 1 and ∆ has the form ∆ = [−b, a] ⊂ R, where a, b ≥ 0 and |a + b| 6= 0. Using (3) one obtains Z a eiaλ − e−ibλ . K[−b,a] (λ) = eiλx dx = iλ −b Note, that as λ → 0 it holds |K[−b,a] (λ) | → b + a < ∞. Now, as λ → ∞ iaλ e − e−ibλ eiaλ + e−ibλ 2 ≤ |K[−b,a] (λ) | = = . iλ |iλ| |λ|

18

T. Alodat et al.

Let τ1 = τ2 = · · · = τκ = α. Then, by (10) Iκ (α, . . . , α) can be estimated as Z Z |K[−b,a] (λ) |2 dλ dλ ≤ C Iκ (α, . . . , α) ≤ CI1 (κα) = C 1 1−κα−2β 1−κα−2β |λ| |λ| |λ|≤C0 R Z C0 Z ∞ Z dρ dρ dλ + C2 ≤C +C . (15) 3−κα−2β 1−κα−2β 3−κα−2β ρ 0 C0 ρ |λ|>C0 |λ| Note, that the two conditions 1 − κα − 2β < 1 and 3 − κα − 2β > 1 are required to guarantee that, integrals in (15) are finite. The first condition κα imples 1 − κα 2 − β < 1 and the second one 1 − 2 − β > 0. So Iκ (α, . . . , α) < ∞ if H ∈ (0, 1). Example 2 Let ∆ be an n-dimensional ball of radius 1, i.e. ∆ = v(1) ⊂ Rn . In this case Z eihλ,xi dx. Kv(1) (λ) = v(1)

Note, that as kλk → 0 it holds |Kv(1) (λ) | ≤ C < ∞. Now, as kλk → ∞ we obtain Jn/2 (kλk) C < |Kv(1) (λ) | = C . kλkn/2 kλk n+1 2

Let τ1 = τ2 = · · · = τκ = α. Then, by (10) Iκ (α, . . . , α) can be estimated as Z |Kv(1) (λ) |2 dλ Iκ (α, . . . , α) ≤ C kλkn−κα−2β n ZR Z dλ dλ ≤ C1 + C2 n−κα−2β 2n−κα−2β+1 kλk kλk kλk≤C0 kλk>C0 # "Z Z ∞ C0 dρ dρ . (16) + ≤C n−κα−2β+2 ρ1−κα−2β C0 ρ 0 The two integrals in (16) are finite provided that 1 − κα − 2β < 1 and β 1 1 1 < 1− κα n−κα−2β +2 > 1. It follows that 12 − 2n 2n − n < 1, i.e. H ∈ ( 2 − 2n , 1), where n ∈ N. Note, that for n = 1 the Hurst index H ∈ (0, 1) and one obtains 1 < 21 . the same result as in Example 1. For n > 1 we get γ(v(1)) = 12 − 2n Example 3 Let n = 2, ∆ = ✷(1) = [−1, 1]2 ⊂ R2 . In this case Z 1Z 1 sin λ1 sin λ2 K✷(1) (λ) = K✷(1) (λ1 , λ2 ) = ei(λ1 x1 +λ2 x2 ) dx1 dx2 = . λ1 λ2 −1 −1 Note, that |K✷(1) (λ1 , λ2 ) | ≤ C. When min(λ1 , λ2 ) > C0 > 0 we get |K✷(1) (λ1 , λ2 ) | ≤

C , |λ1 ||λ2 |

Limit theorems for filtered long-range dependent random fields

19

and if λj > C0 > 0, λi ≤ C0 , i, j ∈ {1, 2}, i 6= j, then sup |K✷(1) (λ1 , λ2 ) | ≤

λi ,i6=j

C . |λj |

(17)

Let us split R2 into the regions A′1 := {(λ1 , λ2 ) ∈ R2 : |λ1 | ≤ C0 , |λ2 | ≤ C0 },

A′2 := {(λ1 , λ2 ) ∈ R2 : |λ1 | ≤ C0 , |λ2 | > C0 },

A′3 := {(λ1 , λ2 ) ∈ R2 : |λ1 | > C0 , |λ2 | ≤ C0 },

A′4 := {(λ1 , λ2 ) ∈ R2 : |λ1 | > C0 , |λ2 | > C0 }, where C0 > 0. Then I1 (κα) can be written as I1 (κα) = (j)

where I1 (κα) :=

|K✷(1) (λ)|2 dλ A′j kλk2−κα−2β ,

R

4 X j=1

(j)

I1 (κα),

(18)

j = 1, . . . , 4. (1)

We will consider each term in (18) separately. The term I1 (·) can be estimated as Z Z |K✷(1) (λ) |2 dλ dλ1 dλ2 (1) I1 (κα) = ≤ C 1− κα 2−κα−2β 2 −β A′1 kλk A′1 (|λ1 ||λ2 |) !2 Z dλ1 . ≤C 1− κα 2 −β |λ1 |≤C0 |λ1 | The last integral is finite provided that 1 − κα 2 − β < 1, i.e. H < 1. (2) Using (17) the term I1 (·) can be estimated as Z Z |K✷(1) (λ) |2 dλ dλ1 dλ2 (2) I1 (κα) = ≤ C 1− κα 2−κα−2β 2 2 −β ′ ′ kλk A2 A |λ2 | (|λ1 ||λ2 |) Z Z 2 dλ1 dλ2 . ≤C κα κα 1− 3− −β 2 2 −β |λ1 |≤C0 |λ1 | |λ2 |>C0 |λ2 | κα The last integrals are finite provided that 1 − κα 2 − β < 1 and 3 − 2 − β > 1. It (3) follows that H ∈ (0, 1). Similarly, one obtains I1 (κα) < ∞ when H ∈ (0, 1). (4) Now, for the term I1 (·) we obtain Z Z |K✷(1) (λ) |2 dλ dλ1 dλ2 (4) I1 (κα) = ≤C 1− κα 2−κα−2β 2 2 2 −β ′ ′ kλk A4 A4 |λ1 | |λ2 | (|λ1 ||λ2 |) ! 2 Z dλ2 . ≤C 3− κα 2 −β |λ2 |>C0 |λ2 |

20

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The last integral is finite provided that 3 − κα 2 − β > 1. It follows that H > 0. By combining the above results for (18), one obtains I1 (κα) < ∞. Therefore, using τ1 = τ2 = · · · = τκ = α and the inequality (10) we obtain that the result of Theorem 7 is true when H ∈ (0, 1). Acknowledgements This research was partially supported under the Australian Research Council’s Discovery Projects (project DP160101366). References 1. Albeverio, S., Molchanov, S.A., Surgailis, D.: Stratified structure of the universe and Burgers’ equation: A probabilistic approach. Probab. Theory Relat. Fields. 100(4), 457– 484 (1994) 2. Alodat, T., Olenko, A.: Weak convergence of weighted additive functionals of long-range dependent fields. Theory Probab. Math. Statist. 97, 9–23 (2017) 3. Alomari, H.M., Ayache, A., Fradon, M., Olenko, A.: Estimation of seasonal long-memory parameters. arXiv preprint arXiv:1805.11905 (2018) 4. Anh, V., Leonenko, N., Olenko, A.: On the rate of convergence to Rosenblatt-type distribution. J. Math. Anal. Appl. 425(1), 111–132 (2015) 5. Anh, V., Leonenko, N., Olenko, A., Vaskovych, V.: On rate of convergence in non-central limit theorems. will appear in Bernoulli. arXiv preprint arXiv:1703.05900 (2017) 6. Bai, S., Taqqu, M.S.: Multivariate limit theorems in the context of long-range dependence. J. Time Series Anal. 34(6), 717–743 (2013) 7. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1989) 8. Bogachev, V.I.: Measure Theory. Springer Verlag, Berlin (2007) 9. Davydov, Y.A.: The invariance principle for stationary processes. Theory Probab. Appl. 15(3), 487–498 (1970) 10. Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functional of Gaussian fields. Probab. Theory Relat. Fields. 50(1), 27–52 (1979) 11. Doukhan, P., Oppenheim, G., Taqqu, M.S.: Theory and Applications of Long-Range Dependence. Birkhuser, Boston (2002) 12. Eisenbaum, N., Kaspi, H.: A characterization of the infinitely divisible squared Gaussian processes. Ann. Probab. 34(2), 728–742 (2006) ´ 13. Eisenbaum, N., Tudor, C.A.: On squared fractional Brownian motions. In: Emery, M., Ledoux, M., Yor, M. (eds.) S´ eminaire de Probabilit´ es XXXVIII. Lecture Notes in Mathematics, vol. 1857, pp. 282–289. Springer, Berlin (2005) 14. Ivanov, A., Leonenko, N.: Statistical Analysis of Random Fields. Kluwer Academic, Dordrecht (1989) 15. Ivanov, A., Leonenko, N.: Semiparametric analysis of long-range dependence in nonlinear regression. J. Statist. Plann. Inference. 138(6), 1733–1753 (2008) 16. Ivanov, A., Leonenko, N., Ruiz-Medina, M.D., Savich, I.N.: Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra. Ann. Probab. 41(2), 1088–1114 (2013) 17. Jazwinski, A.: Stochastic Processes and Filtering Theory. Academic Press, New York (1970) 18. Kallianpur, G.: Stochastic Filtering Theory. Springer, New York (2013) 19. Kratz, M., Vadlamani, S.: Central limit theorems for Lipschitz–Killing curvatures of excursion sets of Gaussian random fields. J. Theoret. Probab. 31(3), 1729–1758 (2018) 20. Leonenko, N.: Limit Theorems for Random Fields with Singular Spectrum. Kluwer Academic, Dordrecht (1999) 21. Leonenko, N., Olenko, A.: Sojourn measures of Student and Fisher–Snedecor random fields. Bernoulli. 20(3), 1454–1483 (2014) 22. Leonenko, N., Ruiz-Medina, M.D., Taqqu, M.S.: Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence. Stoch. Anal. Appl. 35(1), 144– 177 (2017)

Limit theorems for filtered long-range dependent random fields

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