ASYMPTOTICS FOR LINEAR RANDOM FIELDS - UniFI：渐近线性随机域-统一.doc

Asymptotics for Linear Random Fields Research supported by MURST. This is a shorter version of another paper with the same title, to be submitted for publication elsewhere.Teoria Asintotica per Campi Aleatori LineariDomenico MarinucciUniversit La Sapienza, via del Castro Laurenziano 9, 00161 Romaemail: marinuccscec.eco.uniroma1.itSuren PoghosyanInstitute of Mathematics, Armenian National Academy of Sciences, ArmeniaRiassunto: Dimostriamo che le somme parziali di processi stocastici lineari a pi parametri possono essere rappresentate come somme parziali di innovazioni indipendenti pi componenti che sono uniformemente di ordine inferiore. Questa rappresentazione viene utilizzata per stabilire un teorema del limite centrale funzionale ed approssimazioni quasi certe per campi aleatori.Keywords: Approximation of linear random fields, invariance principles, Hungarian construction1. IntroductionIn this paper, we are interested in functional central limit theorems (invariance principles) and almost sure approximations for linear random fields u(t1, tp) on the lattice Zp, p1. Invariance principles for mixing and martingale-difference fields were presented by Goldie and Greenwood (1986), Chen (1991) and Poghosyan and Roelly (1998); while Dedecker (1998) is concerned with central limit theorems. Strong approximations for random field when the u(t1, tp) are independent are provided by Rio (1993). Our purpose is to establish an invariance principle and strong approximations under linear conditions, which are not covered by any of the previous assumptions.The main idea of the paper is to apply to random fields martingale-type approximations which are widely adopted for random sequences, see for instance Marinucci and Robinson (1999). More precisely, we extend to p1 a technique developed by Phillips and Solo (1992) for single-parameter stochastic processes, and we exploit it to decompose partial sums of linear fields into a partial sum of independent components and a remainder, which is shown to be uniformly of smaller order on Zp. Among statistical applications, we mention testing for homogeneity for a character sampled in two different regions, assuming the border of these regions is unknown (i.e. its determination is endogenous to the problem); the asymptotic properties of any procedure based on partial sums will eventually follow from the results presented here and the continuous mapping theorem, much as it happens when testing for structural breaks of unknown location in a time series framework. The plan of this paper is as follows: in Section 2 we establish a decomposition of multivariate polynomials which is the main tool for the subsequent arguments. In Sections 3 and 4 we present the application of this result and we establish invariance principles and almost sure approximations for random fields. Throughout the paper, we use C to denote a positive constant which may vary from line to line and . to denote the integer part of a real number.2. Decomposition of Multivariate PolynomialsConsider the multivariate polynomial A(x1, xp)= i k a(i,k)(x1)i(xp)k , where we assume that |xi|1, i=1,.,p, and Assumption A i k i.k|a(i,k)| .Assumption A is a mild summability condition, which is implied for instance by |a(i,k)| 0 .The following Lemma generalizes a result given for p=1 by Phillips and Solo (1992).Lemma 2.1 Let Gp be the class of all subsets g of (1,2,p). Let yj=xj if jg and yj=1 if jg; we have A(x1, xp)=g jg (xj-1) Ag(y1, yp) ,where products and sums over empty sets are taken to be zero and A(y1, yp)= i k ag(i,k)(y1)i(yp)k , ag(i,k)= i k a(s1,sp)and the sums go over indexes sj, jgj.Proof See Marinucci and Poghosyan (1999).Example Let A(x,y)=1+x+xy+y2;then for g=,(1),(2),(1,2) we have, respectively A(1,1)=4 , A1(x,1)=2 , A2(1,y)=2+y , A12(x,y)=1 ,and hence A(x,y)=4+(x-1)2+(y-1)(2+y)+(x-1)(y-1) .3. The Functional Central Limit TheoremWe are interested in this section in invariance principles for random fields (multiparameter stochastic processes), i.e. array of random variables defined on Rp and taking values on R; more precisely, we introduce the following Assumption B The random variables x(t1,tp) are independent and identically distributed with Ex(t1,tp)=0, Ex(t1,tp)2=s20 and E|x(t1,tp)|q2p; also we have u(t1,tp)= i k a(i,k)x(t1-i,tp-k) .The class of processes defined by Assumptions B covers for instance all stationary Gaussian random fields on Zp, provided a multivariate Wold decomposition is feasible; a sufficient condition for a stationary random field to admit a Wold decomposition is that the log of its spectral density be integrable, see Guyon (1995).Now let W(.,.) denote multiparameter standard Brownian motion, i.e a zero-mean Gaussian process with covariance function satisfying EW(t1,tp)W(s1,sp)=jmin(tj,sj) ;also, let Dp be the space of “cadlag” functions form 0,1p to R; it is possible to introduce on Dp a metric topology which makes it complete and separable, and indeed Dp is the multidimensional analogue of the Skorohod space D0,1, see Poghosyan and Roelly (1998) for details. Now let i denote the sum for ti=1,2,nri, where 1ri0; the ideas of Section 2 can be exploited to prove the following Theorem, which is the main result of this paper.Theorem 3.1 Under Assumptions A and B, as n, sn-p/2 1p u(t1,tp)A(1,1) W(r1,rp) ,where signifies weak convergence in Dp.Proof See Marinucci and Poghosyan (1998).4. The Hungarian ConstructionIn the last decades, a considerable amount of effort has been devoted to the investigation of the possibility to approximate almost surely partial sums of i.i.d. random variables with partial sums of i.i.d. Gaussian variables. Such results are usually referred to as Hungarian constructions and they have proved to be extremely valuable tools in a number of ares of probability and mathematical statistics. The following lemma provides a special case of a result established by Rio (1993).Lemma 4.1 (Rio (1993) Let x(t1,tp) be a random field of independent variables with Ex(t1,tp)=0, Ex(t1,tp)2=s20 and E|x(t1,tp)|q2p/(p-d), some d0. Then there exist a Gaussian random field z(t1,tp) of zero-mean independent variables with variance s2 such that, for all (r1,rp) in 0,1p, 1d0, pd, as n |i x(ir1,irp)- z(ir1,irp)|=O(n(p-d)/2(logn)1/2) a.s. .Our purpose in this Section is to generalize Lemma 4.1 to the case of dependent arrays of random variables:Theorem 4.1 Let u(t1,tp) be a random field of identically distributed variables such that Assumptions A and B hold, the latter strengthened to q2/(1-d), 0d1. Then there exist a Gaussian random field z(t1,tp) of zero-mean independent variables with variance A(1,1)2s2 such that, for all (r1,rp) in 0,1p, as n |i x(ir1,irp)- z(ir1,irp)|=O(n(p-d)/2(logn)1/2) a.s. .Proof See Marinucci and Poghosyan (1999).To the best of our knowledge, Theorem 4.1 is the first result establishing strong approximations for random fields when the assumption of independence is relaxed.ReferencesChen, D. (1991) “A uniform central limit theorem for nonuniform f-mixing random fields”, Annals of Probability, 19, 636-649Dedecker, J. (1998) “A central limit theorem for stationary random fields”, Probability Theory and Related Fields, 110, 397-426Goldie, C.M. and Greenwood, P.E. (1986) “Variance