Asymptotics for Linear Random Fields1
Teoria Asintotica per Campi Aleatori Lineari
Domenico Marinucci
Università La Sapienza, via del Castro Laurenziano 9, 00161 Roma
email: marinucc@scec.eco.uniroma1.it
Suren Poghosyan
Institute of Mathematics, Armenian National Academy of Sciences, Armenia
Riassunto: Dimostriamo che le somme parziali di processi stocastici lineari a più
parametri possono essere rappresentate come somme parziali di innovazioni
indipendenti più 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
construction
1. Introduction
In 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, p>1. 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 p>1 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
1
Research supported by MURST. This is a shorter version of another paper with the same title, to be
submitted for publication elsewhere.
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 Polynomials
Consider 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)| <C( i..k)2- , some >0 .
The following Lemma generalizes a result given for p=1 by Phillips and Solo (1992).
Lemma 2.1 Let p be the class of all subsets  of (1,2,…,p). Let yj=xj if j and yj=1
if j; we have
A(x1,…, xp)= j (xj-1) A(y1,…, yp) ,
where products and sums over empty sets are taken to be zero and
A(y1,…, yp)= i …k a(i,…,k)(y1)i…(yp)k ,
a(i,…,k)= i …k a(s1,…,sp)
and the sums go over indexes sj, jj.
Proof See Marinucci and Poghosyan (1999).
Example Let
A(x,y)=1+x+xy+y2;
then for =,(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 Theorem
We 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 (t1,…,tp) are independent and identically
distributed with E(t1,…,tp)=0, E(t1,…,tp)2=2>0 and E|(t1,…,tp)|q< , q>2p; also
we have
u(t1,…,tp)= i …k a(i,…,k)(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 [0,1] p 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 D[0,1], see Poghosyan and
Roelly (1998) for details. Now let i denote the sum for ti=1,2,…,[nri], where 1ri0;
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,
n-p/2 1…p u(t1,…,tp)A(1,…,1) W(r1,…,rp) ,
where  signifies weak convergence in Dp.
Proof See Marinucci and Poghosyan (1998).
4. The Hungarian Construction
In 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 (t1,…,tp) be a random field of independent variables with
E(t1,…,tp)=0, E(t1,…,tp)2=2>0 and E|(t1,…,tp)|q< , q>2p/(p-), some >0. Then
there exist a Gaussian random field (t1,…,tp) of zero-mean independent variables with
variance 2 such that, for all (r1,…,rp) in [0,1] p, 1>0, p>d, as n
|i ([ir1],…,[irp])- ([ir1],…,[irp])|=O(n(p-)/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 q>2/(1-), 0<<1. Then
there exist a Gaussian random field (t1,…,tp) of zero-mean independent variables with
variance A(1,…,1)22 such that, for all (r1,…,rp) in [0,1] p,, as n
|i ([ir1],…,[irp])- ([ir1],…,[irp])|=O(n(p-)/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.
References
Chen, D. (1991) “A uniform central limit theorem for nonuniform -mixing random
fields”, Annals of Probability, 19, 636-649
Dedecker, J. (1998) “A central limit theorem for stationary random fields”, Probability
Theory and Related Fields, 110, 397-426
Goldie, C.M. and Greenwood, P.E. (1986) “Variance of set-indexed sums of mixing
random variables and weak convergence of set-indexed processes”, Annals of
Probability, 14, 817-839
Guyon, X. (1995) Random Fields on a Network. Modeling, Statistics and Applications.
Springer, Verlag
Marinucci, D. and Poghosyan, S. (1999) “Asymptotics for linear random fields”,
Working Papers Ricerca Nazionale Modelli Statistici per l’Analisi delle Serie
Temporali, n.5
Marinucci, D. and Robinson, P.M. (1999) “Weak convergence of multivariate
fractional processes”, Stochastic Processes and their Applications, in press
Phillips, P.C.B. and Solo, V. (1992) “Asymptotics for linear processes”, Annals of
Statistics, 20, 971-1001
Poghosyan, S. and Roelly, S. (1998) “Invariance principle for martingale-difference
random fields”, Statistics and Probability Letters, 38, 235-245