RATE OF CONVERGENCE TO THE ROSENBLATT DISTRIBUTION FOR ADDITIVE FUNCTIONALS OF STOCHASTIC PROCESSES WITH LONG-RANGE DEPENDENCE

This paper establishes the rate of convergence (in the uniform Kolmogorov 
distance) for normalized additive functionals of stochastic processes with 
long-range dependence to a limiting Rosenblatt distribution.


Introduction
The study of random processes and fields with correlations decaying at hyperbolic rates, i.e., those with long-range dependence (LRD), presents interesting and challeng- ing probabilistic as well as statistical problems.Progress has been made in the past two decades or so on the theoretical aspects of the subject.On the other hand, recent applications have confirmed that data in a large number of fields (including hydrolo- gy, geophysics, turbulence, economics and finance) display LRD.Many stochastic models have been developed for description and analysis of this phenomenon.For re- cent developments, see Beren [6], Barndorff-Nielsen [5], Anh and Heyde [2], Leonenko 1The work was partially supported by the Australian Research Council grant A69804041 and NATO grant PST.CLG.976361.
The non-central limit theorem, which describes the limiting distributions of addi- tive functionals, plays a key role in the theory of random processes and fields with LRD.The main references here are Taqqu [35,37], and Dobrushin and Major [10] (see also, Surgailis [34], Samorodnitsky and Taqqu [33], Leonenko and ilac-Benic [21], Ho and Hsing [16], Leonenko and Woyczynski  [24], Ash and Leonenko [4], among others).The limiting distributions have finite second-order moment but may have non-Gaussian structure.The problem of rate of convergence in the non-central limit theorem is therefore of considerable interest.Some results on the rate of convergence to the Gaussian distribution for integral functionals of Gaussian random processes and fields with LRD were considered by Leonenko  [19] (see also, Ivanov and Leonenko [17], pp. 64-70,Leonenko et al. [22], and Leonenko and Woyczynski [23]).These results correspond to Hermitian rank m 1 (defined in Section 2).In this paper, we provide the rate of convergence (in the uniform Kolmogorov dis- tance) of probability distributions of normalized integral functionals of Gaussian pro- cesses with LRD and a special form of the covariance function (see condition C below) to a limiting non-Gaussian distribution called the Rosenblatt distribution.
The function Bo(t), t E , is known as the Fourier transform of Bessel potential (see Donoghue  [11], p. 294) or characteristic function of symmetric Bessel distributions (see Oberhettinger [27], p. 156, or Fang et al. [14], p. 69).It has the spectral repre- sentation Bo(t ] cos(At)f(A)d, (2) with an exact form of spectral density fa (see (17) below)such that fa(,) Cl(OZ) a-1 0<a<l as ---0, where the Tauberian constant c(c) 2r(c)cols(cr/2). (4) The process (t), t E R, itself satisfying condition A with covariance function B0, has the spectral representation (t) / eitv/f()W(dA), (5) where W(. is the complex Gaussian white noise. The function Bl(t), t G N, is known as the characteristic function of the Linnik dis- tribution (see Kotz et al. [18]).This distribution has a density function (i.e., the co- variance function B 1 has a spectral density).Kotz et al. [18] investigated the asymp- totic behavior at frequency 0 are quite distinct in the cases: (i) 1/a being an integer, (ii) 1/a being a non-integer rational number, and (iii) a being an irrational number.Similar properties hold for the covariance function B2(t), t , which is known as the characteristic function of the generalized Linnik distribution (see Erdoan and Ostrovskii [13]).
In this paper, we shall consider the covariance function Bo(t), t , as the key example of covariance functions of random processes in continuous time in the sense of representation (1).In principle, our method is also applicable to the cases of covar- iance functions Bl(t and B2(t).The method uses the second term in the asymptotic expansion of the spectral density at frequency zero, which depends on the arithmetic nature of the parameter c (see Kotz et al. [18]).Hence the rate of convergence de- pends on the arithmetic nature of a and is different for the cases (i)-(iii).This pro- blem will be addressed elsewhere.
Viano et al. [39] introduced continuous-time fractional ARMA processes.Some asymptotic results for the correlation functions and spectral densities of these process- es were obtained.However, these results are not useful to the problem of this paper, since in our approach we need exact results (such as Lemma 4.5 below) on the asymp- totic behavior of the spectral density at frequency zero.See also Remark 3.3 below.B.
Additionally, we will assume that the function G satisfies the condition B'.
Remark 2.1: The definition and properties of the multiple stochastic integral (8) can be found in Major [25] or Taqqu [37].
For a random process in continuous time, the proof of Theorem 2.1 may be con- structed from Taqqu [37] and Dobrushin and Major [15] by using the argument of Berman [7].
The Gaussian process Yl(S), s > 0, defined in (8) with m 1, is fractional Brown- ian motion.This process plays an important role in applications in hydrology, turbu- lence, finance, etc.An extension of this process has been recently proposed by Anh et   al. [3].They introduced fractional Riesz-Bessel motion, which provides a generaliza- tion of fractional Brownian motion and describes long-range dependence as well as second-order intermittency.The latter is another important feature of turbulence and financial processes.The spectral density of increments of such processes is a generalization of the spectral density of fractional Ornstein-Uhlenbeck-type processes (see Comte [9]).
It is easy to see that t is a compact operator and the bounds are different from zero; therefore they are in the spectrum of t.Thus, there exist at least two non-zero eigenvalues V p and Uq such that /p (16) In fact, at least one non-zero eigenvalue exists because t is a non-zero operator with non-zero norm.
Suppose that there is only one non-zero eigenvalue u with corresponding non-zero eigenvector bl(A); then putting "1 A2 in (15) and using (11)   we obtain A 1 $1(A1) 2 0, which is a contradiction.Using the same argument, it is easy to prove that if there exist two non-zero eigenvalues, then they are different.
Recently, Albin [1] proved that the Rosenblatt distribution has a density function which belongs to the type-1 domain of attraction of extremes.Albin [1] also used the representation (13) where and the Laplace transform of R 2 is given by Eexp{-sR2}-exp ln(1 )

Main Result
We present a result on the rate of convergence in the uniform Kolmogorov distance) of probability distributions of random variables YT(1), defined in (7) for a special covariance function (see condition C below), to the Rosenblatt distribution of R2(1), defined in (10) or ( 12) and (13).Some results on the rate of convergence to the normal law along the line of Theorem 2.1 were obtained by Leonenko  [19] (see also Ivanov and Leonenko [17, p. 64-70]).These results correspond to the case rn 1 (see condition B') in Theorem 2.1.In this paper, we examine the case m 2.
For technical reasons, we formulate the following assumption for the covariance function. C.
Introduce a uniform (or Kolmogorov's) distance between the distributions of the random variables X and Y via the formula %(2, Y) sup lP(X <_ z)-P(Y 5 The main result of this paper describes the rate of convergence (as T+c) in Theorem 2.1 with m-2 and is contained in the following.
Theorem 3.1: Let assumptions A, B, B' and C hold with m-2 and c @ (0,1/3).where Co, C 2 are defined in (6) and the constant Cl(C is defined in (4).The numbers p and ,q are defined in (16), and the random variable R2(1 which has the Rosenblatt distribution is defined in (10) or ( 12) and (13).
Remark 3.3: Our methodology in principle, is applicable to more general Gaussian processes in continuous time.For this, we have to replace condition C by a more general condition which can be given in the spectral form, such as (21), together with the type of results of Lemma 4.5 and a precise behavior of the spectral density near infinity (for example, f(A) O(]A[ -1-a) as ])[--OO).Then, instead of the convolution property (22), we may use the Riesz Composition Formula (see Lemma 4.6 below) for an investigation of the asymptotic behavior of convolutions of spectral density.Lemma 4.2 can next be used again to obtain the asymptotic formulae similar to (9) but in terms of the spectral density.Then the proof can be completed by following the same principal steps.We will address this approach in a separate paper together with a generalization to random fields.

Concluding Remarks
This paper addresses the issue of measuring the speed of convergence to the Rosenblatt distribution, as measured by the Kolmogorov distance, for some functionals of nonlinear transformations of long-range dependent Gaussian processes with Hermite rank m 2. Our method is based on a direct probabilistic analysis of the main term (m 2) as well as the second term (m 3).Due to the nature of limiting laws in the situation of LRD, it is not straightforward to present an argument on the sharp- ness of the results as in the traditional situation of short-range dependence.In parti- cular, the rate of convergence in Theorem 5.1 is not optimal, hence yields a gap in the rate of convergence at a 1/3 between Theorems 3.1 and 5.1.However, the paper takes the first step towards solving the important and difficult problem of sharp convergence rate in non-central limit theorems.The method of this paper in fact is general.It can be applied to nonlinear functionals of non-Gaussian random processes with LRD and special bilinear expansions of their bivariate densities in orthogonal polynomials such as Chebyshev- Hermite polynomials, Laguerre polynomials.In particular, the rate of convergence to the non-Gaussian Laguerre processes with Laguerre rank rn 1 has been obtained in Anh and Leonenko [4] (see also Leonenko [20]).

2 2
is of the form density function of the random variable t3Xp + X q v()-/ v (,-x)Vx(

gl
Some further information on the density function of the Rosenblatt distribution can be found in Albin[1].