The Multivariate Empirical of Long Memory Processes

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 be a d-variate linear process independent of the form:

(1)

Given the set of observations (X₁₁,..., X_1n),...,(X_n1,..., X_nn), let be the empirical marginal distribution function, where 1A denotes the indication function of set A; we can then introduce the multivariate empirical process for , a normalizing factor to be discussed later.

The asymptotics for G_n(x₁,...,x_d) 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 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 proofs of intermediary results are collected in the appendix.

Assumptions and Motivations

Our first condition relates to some unobservable sequences ε_t1,...,ε_td, 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

(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 are positive slowly varying functions [8].

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

The observable sequences are subordinated to εt in the following sense.

Condition B.

For some real, measurable deterministic functions

We stress that we are imposing no restriction other than measurability on 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 , 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 H_p(.) the p-th order Hermite polynomial, the first few being,

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

for and 0 if not.

Hence, under condition A,

where

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

We can expand into orthogonal components, as follows:

(3)

where the coefficients are obtained by the standard projection formula

From (3) we have, for any fixed ,

(4)

It is thus intuitive that the stochastic order of magnitude of is determined by the lowest terms corresponding to non-zero such that,

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

Condition C.

Condition C entails that the covariances of are not summable, i.e. they display long memory behaviour.

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,

be the square root of the asymptotic variance of , we need the following technical condition.

Condition D.

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

Of course, we have

Thus, condition D is a mild regularity assumption on the slowly varying functions .

Main Result

Define the random processes

where W₁(.)...W_d(.) are independent copies of a Gaussian white noise measure on , the integrals exclude the hyper diagonals, and

(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,

 

(7)

where denotes weak convergence in the Skorohod space . We provide now a uniform reduction principle for the multivariate case.

Proposition 2

Under conditions A, B, C and D,

Theorem

Under conditions A, B, C and D,

where denotes weak convergence in the Skorohod space.

Appendix

Proof of Proposition 1

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

(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 ε_tj can be given a spectral representation as

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

With governing spectral measures:

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

Next we de ne new random measures on the Borel sets by

so that after the change of variables for equation (8) becomes:

Now consider the spectral measures,

and a piecewise constant modification of the Fourier transform, i.e.

Where the last step follows from

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

Lemma 1.A

As we have, uniformly in every bounded region

Where

Proof

Let

it can be verified that

Now define the set

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

Where

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

(9)

(10)

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

Choose a positive small enough that

Then

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

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

and such that

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

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

Lemma 2. A

Let G_jn be sequences of non-atomic spectral measures on B on tending locally weakly to d non-atomic spectral measures G_j0, j = 1,…, d, Kn(ε₁,… ε_pd) a sequence of measurable functions on tending to a continuous functionin any rectangle Let the functions Kn(.) satisfy the relation

(11)

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

exists, and as

where Z_Gj0(.) denotes a random to be dened below, and based on G_j0(.), 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, Z_G1n(.) ... Z_Gdn(.); 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.

and

The convergence of Kn (.) to K (.) in any rectangle is immediate.

The convergence of the measures G_jn (.) to G_j0 (.), 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

and

Note that is the Fourier transform of and is the Fourier transform of . By lemma 1.A,converges to uniformly in every bounded region, and hence by lemma 2 in DM [4] we have that tends weakly to the measure ,which must be finite. Moreover, weak convergence entails that

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

(12)

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

(13)

Now note that, for any belongs to by 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

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

It is obvious that, if x_1i,...,x_dl, for i = 1 ,....,I, and l = 1_i,....,L, are no decreasing sequences, then the sets are all disjoint. Given any multivariate function we can hence define an associated (signed) measure by,

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

(14)

Proof

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

because

for all (p₁...p_d) such that .

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

Set

and

it can be readily verified that, for any give block

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

The sequences become finer and finer as μ_j and j = 1,...,d grow, i.e.

.

Clearly, we have

For the following, we put i = j_iand j = j_d.

Now consider the sets

which define a net of refining partitions of , i.e.

Note also that

Define by

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

(15)

(16)

(17)

(18)

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

Therefore,

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

Since for any

we have

.

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

And hence

(19)

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

The remaining part of the argument is entirely an analogous

.

.

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

and thus the result is established.

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