The Multivariate Empirical of Long Memory Processes | Open Access Journals

The Multivariate Empirical of Long Memory Processes

Ichaou Mounirou*

Faculty of Management and Economics Sciences, Université de Parakou, Benin

*Corresponding Author:
Mounirou Ichaou
Faculty of Management and Economics Sciences, Université de Parakou, Benin
E-mail: ichaou_bassir@yahoo.fr

Received date: 10/03/16; Accepted date: 14/04/16; Published date: 18/04/16

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Abstract

We establish a functional central limit theorem for the empirical pro-cess of long range dependent stationary multivariate sequences under Gaussian subordination conditions. The proof is based upon a convergence result for cross-products of Hermite polynomials and a multivariate uniform reduction principle, as in Marinucci for bivariate sequences.

Keywords

Multivariate processes, Empirical process, Hermite polynomials, Convergence

Introduction

Let statistics-and-mathematical-sciences be a d-variate linear process independent of the form:

statistics-and-mathematical-sciences (1)

Given the set of observations (X11,..., X1n),...,(Xn1,..., Xnn), let statistics-and-mathematical-sciences be the empirical marginal distribution function, where 1A denotes the indication function of set A; we can then introduce the multivariate empirical process for statistics-and-mathematical-sciences , a normalizing factor to be discussed later.

The asymptotics for Gn(x1,...,xd) when the observables are independent and identically distributed (i.i.d.) or weakly dependent has long been well understood by Dudley [1] for a review. In this paper, we shall focus instead on the case where Xt is a long memory process, in a sense to be rigorously defined in section 2, Marinucci [2] developed in the bivariate case. Our work can hence be seen as an extension to the multivariate case of bivariate results from Marinucci [2]; see also Arcones [3] for results in the multivariate Gaussian case.

The structure of this paper is as follow. In section 2, we introduce our main assumptions and we discuss Hilbert space techniques for the analysis of multivariate long memory processes. Section 3 presents first a convergence result for the finite dimensional distributions of statistics-and-mathematical-sciences the limiting elds can be viewed as straightforward extensions of the Hermite processes considered by Dobrushin and Major [4], Taqqu [5] and many subsequent authors. We then go on to establish a multivariate uniform reduction principale, which extends Dehling and Taqqu [6] and is instrumental for the main result of the paper, i.e. a functional central limit theorem for statistics-and-mathematical-sciences proofs of intermediary results are collected in the appendix.

statistics-and-mathematical-sciences

Assumptions and Motivations

Our first condition relates to some unobservable sequences εt1,...,εtd, which we shall use as building blocks for the processes of interest.

Condition A. The sequences {εt, t = 1,...} are jointly both Gaussian and independent, with zero mean, unit variance and auto covariance functions satisfying, for statistics-and-mathematical-sciences

statistics-and-mathematical-sciences (1)

Condition A.

It is a characterization of regular long memory behaviour, entailing that εt have non-summable autocovariance functions and a spectral density with a singularity at frequency zero (see for instance, Leipus and Viano [7] for a more general characterization of long memory). Here, ∼ denotes that the ratio between the right and left-hand sides tends to one, and statistics-and-mathematical-sciences are positive slowly varying functions [8].

statistics-and-mathematical-sciences , for all c > 0 and La (.) is integrable on every nite interval.

The observable sequences statistics-and-mathematical-sciences are subordinated to εt in the following sense.

Condition B.

For some real, measurable deterministic functions

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

We stress that we are imposing no restriction other than measurability on statistics-and-mathematical-sciences for i = 1,...,d, and consequently condition B covers a very broad range of marginal distributions on Xt; in particular, although Xt are strictly stationary they need not have nite variances and hence be wide sense stationary. If we denote by statistics-and-mathematical-sciences, the cumulative distribution function of a standard Gaussian variate. As in many previous contributions, our idea in this paper is to expand the multivariate empirical process into orthogonal components, such that only a nite number of them will be non-negligible asymptotically. Our presentation will follow the notation by Marinucci. Denote by Hp(.) the p-th order Hermite polynomial, the first few being,

statistics-and-mathematical-sciences

It is known that these functions form a complete orthogonal system in the Hilbert space statistics-and-mathematical-sciencesdenoting a standard Gaussian density. Also, for zero-mean, unit variance variables statistics-and-mathematical-sciences with Gaussian joint distribution we have,

statistics-and-mathematical-sciences

statistics-and-mathematical-sciencesfor statistics-and-mathematical-sciences and 0 if not.

Hence, under condition A,

statistics-and-mathematical-sciences

where

statistics-and-mathematical-sciences

In view of (1) and (2), and using the same argument as in Taqqu [9], theorem 3.1, and Marinucci [2]; here

statistics-and-mathematical-sciences

We can expand statistics-and-mathematical-sciences into orthogonal components, as follows:

statistics-and-mathematical-sciences(3)

where the coefficients statistics-and-mathematical-sciences are obtained by the standard projection formula

statistics-and-mathematical-sciences

From (3) we have, for any fixed statistics-and-mathematical-sciences,

statistics-and-mathematical-sciences(4)

It is thus intuitive that the stochastic order of magnitude of statistics-and-mathematical-sciences is determined by the lowest statistics-and-mathematical-sciencesterms corresponding to non-zero such that,

statistics-and-mathematical-sciences

In the sequel, it should be kept in mind that the cardinality of statistics-and-mathematical-sciences (which we denoted h) can be larger than unity, i.e. the minimum of statistics-and-mathematical-sciences can be non-unique; of course,

statistics-and-mathematical-sciences

Condition C.

Condition C entails that the covariances of statistics-and-mathematical-sciences are not summable, i.e. they display long memory behaviour.

statistics-and-mathematical-sciences

Note that for condition C to hold it is not necessary that the observables X1,...,Xd are long memory; the autocovariances of one of them can be summable.

Now let,

statistics-and-mathematical-sciences

be the square root of the asymptotic variance of statistics-and-mathematical-sciences, we need the following technical condition.

Condition D.

As statistics-and-mathematical-sciences exists and it is non-zero, i.e. there exist some positive, finite constants statistics-and-mathematical-sciences such that

statistics-and-mathematical-sciences

Of course, we have

statistics-and-mathematical-sciences

Thus, condition D is a mild regularity assumption on the slowly varying functions statistics-and-mathematical-sciences .

Main Result

Define the random processes

statistics-and-mathematical-sciences

where W1(.)...Wd(.) are independent copies of a Gaussian white noise measure on statistics-and-mathematical-sciences, the integrals exclude the hyper diagonals, and

statistics-and-mathematical-sciences(6)

for j = 1,...,d.

Indeed, the following result is a direct extension of results by Marinucci [2]

Proposition 1

Under conditions A, B, C and D, statistics-and-mathematical-sciences

 

statistics-and-mathematical-sciences(7)

where statistics-and-mathematical-sciences denotes weak convergence in the Skorohod space statistics-and-mathematical-sciences. We provide now a uniform reduction principle for the multivariate case.

Proposition 2

Under conditions A, B, C and D, statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

Theorem

Under conditions A, B, C and D, statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

where statistics-and-mathematical-sciences denotes weak convergence in the Skorohod spacestatistics-and-mathematical-sciences.

Appendix

Proof of Proposition 1

In the sequel, we concentrate, for notational simplicity, on the case h = 1 and we write for brevity statistics-and-mathematical-scienceswhen no confusion is possible. We focus first on the asymptotic behavior of

statistics-and-mathematical-sciences (8)

Here our proof is basically the same as the well-known argument by Dobrushin and Major [4] for univariate Hermite polynomials, and Marrinucci [2] for bivariate case, we omit many details. The sequences εtj can be given a spectral representation as

statistics-and-mathematical-sciences

Where, by condition A and Zygmund’s lemma [10]

statistics-and-mathematical-sciences

With governing spectral measures:

statistics-and-mathematical-sciences

Hence, by the well-known formula relating Hermite polynomials to Wiener-Ito integrals [11]

statistics-and-mathematical-sciences

Next we de ne new random measures on the Borel sets statistics-and-mathematical-sciences by

statistics-and-mathematical-sciencesstatistics-and-mathematical-sciences so that after the change of variables statistics-and-mathematical-sciences for statistics-and-mathematical-sciences equation (8) becomes:

statistics-and-mathematical-sciences

Now consider the spectral measures,

statistics-and-mathematical-sciencesand a piecewise constant modification of the Fourier transform, i.e.

statistics-and-mathematical-sciences

Where statistics-and-mathematical-sciencesthe last step follows from

statistics-and-mathematical-sciences

The following result is a simple extension of lemma 1 in DM [4] and lemma A.1 in Marrinucci [2].

Lemma 1.A

As statistics-and-mathematical-sciences we have, uniformly in every bounded region

statistics-and-mathematical-sciences

Where

statistics-and-mathematical-sciences

Proof

Let

statistics-and-mathematical-sciences

it can be verified that

statistics-and-mathematical-sciences

Now define the set

statistics-and-mathematical-sciences

As in DM [4], by the standard properties of slowly varying functions, it can be shown that, for any c,statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

Where

statistics-and-mathematical-sciences

To complete the proof, we just need to show that,

statistics-and-mathematical-sciences(9)

statistics-and-mathematical-sciences (10)

For every l = 1,..., p1 + ... + pd, such that |ul| < c. We assume without loss of generality that p1,...,pd# 0, otherwise we are back to the univariate case.

Choose a positive statistics-and-mathematical-sciences small enough that

statistics-and-mathematical-sciences

Then

statistics-and-mathematical-sciences

Hence by Holder inequality we obtain for equation (10) that

statistics-and-mathematical-sciences

For (9), we can argue exactly as in DM [4], equations (3.9) - (3.10), to show that there must exist α > 0, small enough that

statistics-and-mathematical-sciences

and such that

statistics-and-mathematical-sciences

Then, again as in DM (1979), equation (3.11),we obtain

statistics-and-mathematical-sciences

whence the proof can be completed by the same argument as for (10).

Lemma 2. A

Let Gjn be sequences of non-atomic spectral measures on B on tending locally weakly to d non-atomic spectral measures Gj0, j = 1,…, d, Kn(ε1,… εpd) a sequence of measurable functions on statistics-and-mathematical-sciences tending to a continuous functionstatistics-and-mathematical-sciencesin any rectangle statistics-and-mathematical-sciences Let the statistics-and-mathematical-sciencesfunctions Kn(.) satisfy the relation

statistics-and-mathematical-sciences (11)

uniformly for n = 0, 1….Then the Dobrushin-Wiener-Ito integral

statistics-and-mathematical-sciences

exists, and as statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

where ZGj0(.) denotes a random to be dened below, and based on Gj0(.), j=1,… d

Proof

The proof is identical to the argument by DM (1979, p.41); the définition of local weak convergence is given on page 31. Note that here we have d different random measures, ZG1n(.) ... ZGdn(.); as these d measures are independent, however, the extension to product spaces is straight foward.

To establish the asymptotic behaviour of (8), we apply Lemma 2.A with the choice.

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

and

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

The convergence of Kn (.) to K (.) in any rectangle statistics-and-mathematical-sciencesis immediate.

The convergence of the measures Gjn (.) to Gj0 (.), j = 1,..., d is proved in Proposition 1 by DM [4]. The crucial step is then to show that equation (11) holds.

Consider the d measures

statistics-and-mathematical-sciences

and

statistics-and-mathematical-sciences

Note that statistics-and-mathematical-sciences is the Fourier transform ofstatistics-and-mathematical-sciences and statistics-and-mathematical-sciences is the Fourier transform of statistics-and-mathematical-sciences. By lemma 1.A,statistics-and-mathematical-sciencesconverges to statistics-and-mathematical-sciencesuniformly in every bounded region, and hence by lemma 2 in DM [4] we have that statistics-and-mathematical-sciencestends weakly to the measure statistics-and-mathematical-sciences ,which must be finite. Moreover, weak convergence entails that

statistics-and-mathematical-sciences

(Condition (1.14) in DM [4]), and in turn this implies (11). We have thus shown that, as statistics-and-mathematical-sciences

statistics-and-mathematical-sciences(12)

And also, if we view the left-and right-hand sides of (12) as constant random functions from statistics-and-mathematical-sciences

statistics-and-mathematical-sciences(13)

Now note that, for any statistics-and-mathematical-sciencesbelongs to statistics-and-mathematical-sciencesby its own definition; proposition 1 then follows from the functional versionof Slutsky's lemma and the continuous mapping theorem, see for instance Van Der Vaart and Wellner [12], section 1.4.

Now introduce the function

statistics-and-mathematical-sciences

For the arguments in the sequel, we use the following notation. Let aj ; bj be any uplet of real numbers statistics-and-mathematical-scienceswe can define the blocks

statistics-and-mathematical-sciences

It is obvious that, if x1i,...,xdl, for i = 1 ,....,I, and l = 1i,....,L, are no decreasing sequences, then the sets statistics-and-mathematical-sciencesare all disjoint. Given any multivariate function statistics-and-mathematical-sciences we can hence define an associated (signed) measure by,

statistics-and-mathematical-sciences

The resulting measure can be random, for instance if we take T (;...; ..) = Sn (.;...; .) as we shall often do in the sequel. The following result provides an extension of lemma 3.1 in Dehling and Taqqu [6] to the random measure case.

Lemma 3.A

Under conditions A, B, C and D, there exist some ν > 0 such that, as statistics-and-mathematical-sciences

statistics-and-mathematical-sciences (14)

Proof

With p1...pd = p, in view of equation (3), we obtain

statistics-and-mathematical-sciences

because

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

for all (p1...pd) such that statistics-and-mathematical-sciences.

For notational simplicity and without loss of generality, we consider only the case h = 1; also, we write statistics-and-mathematical-sciences for statistics-and-mathematical-sciences. We use a chaining argument which follows closely the well-known proof of Dehling and Taqqu [6].

Set

statistics-and-mathematical-sciences

and

statistics-and-mathematical-sciences

it can be readily verified that, for any give block statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

The idea is to build a "fundamental" partition of statistics-and-mathematical-sciencesRd, such that statistics-and-mathematical-sciences, for each Δ in this class and for a fixed statistics-and-mathematical-sciences. Starting from this fundamental class, we will then dene coarser partitions by summing blocks made up with statistics-and-mathematical-sciences fundamental elements, μj = 1,2,...,K for j = 1,...,d. The latter blocks will then be used in a chaining argument to establish an uniform approximation of Sn(x1....,xd). More precisely, put

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

The sequences statistics-and-mathematical-sciences become finer and finer as μj and j = 1,...,d grow, i.e.

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences.

Clearly, we have

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

For the following, we put i = jiand j = jd.

Now consider the sets

statistics-and-mathematical-sciences

which define a net of refining partitions of statistics-and-mathematical-sciences, i.e.

statistics-and-mathematical-sciences

Note also that

statistics-and-mathematical-sciences

Define statistics-and-mathematical-sciencesby

statistics-and-mathematical-sciences

And in Marinucci [2], we can use the decomposition

statistics-and-mathematical-sciences(15)

statistics-and-mathematical-sciences(16)

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences(17)

statistics-and-mathematical-sciences(18)

in words, we have partitioned the random measure Sn(x1,...,xd) over 2d sets of blocks: those were the corners are all smaller than x1,...,xd (15), those where the corners have coordinate x2,...,xd−1 and the top corners have coordinate xd (16), those where the right corners have coordinate others variables x1,...,xd−1 (17), and a single block which has (x1,...,xd) as its top right corner (18). Now

statistics-and-mathematical-sciences

Therefore,

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

By an identical argument in Marinucci (2005), finally, we have

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

Since for any statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

we have

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences.

Now note that, by lemma 3.A and Chebyshev's inequality,

statistics-and-mathematical-sciences

And hence

statistics-and-mathematical-sciences

statistics-and-mathematical-sciences(19)

Equation (19) is immediately seen to be o (1). Also, in Marinucci [2], we obtain

statistics-and-mathematical-sciences

The remaining part of the argument is entirely an analogous

statistics-and-mathematical-sciences.

statistics-and-mathematical-sciences.

From the prepositions 1and 2, we have, as n to infinity

statistics-and-mathematical-sciences

statistics-and-mathematical-sciencesand thus the result is established.

References