SOME REMARKS ON THE CENTRAL LIMIT THEOREMFOR FUNCTIONALS OF LINEAR PROCESSES UNDER SHORT-RANGE DEPENDENCE

Sep 15, 2026

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This paper studies central limit theorems for functionals G(Xn) of one-sided m-dimensional linear processes driven by i.i.d. innovations. It assumes short-range dependence and measurable G with finite expectation of |G(X1)|, then derives weak convergence of partial-sum processes to a scaled Brownian motion in the Skorokhod function space. The proof uses martingale decomposition of Yn=G(Xn)-EG(Xn) and applies auxiliary martingale-difference CLTs and weak-convergence results.

LanguageEnglish

CategoryResearch & Report

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Pages11

Reading Time28 min

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1. INTRODUCTION

2. MAIN RESULTS

3. AUXILIARY LEMMAS

# SOME REMARKS ON THE CENTRAL LIMIT THEOREMFOR FUNCTIONALS OF LINEAR PROCESSESUNDER SHORT-RANGE DEPENDENCE BY KONRAD FURMANCZYK(WARSZAWA) Abstract.In this paper we consider the central limit theorems forfunctionals G:R"→R of one-sided m-dimensional linear processes,where A,is a nonrandom matrix m×mand Z's are i.i.d.random vectors in R". 2000 AMS Mathematics Subject Classification:Primary:60F05; Secondary:60G10. Key words and phrases:Central limit theorem,linear process,time series,short-range dependence,Markov chain,martingale diffe-rences. ## 1.INTRODUCTION The one-sided linear process(LP)is defined as follows: j=1,2,…,n, where the innovationsare i.i.d.random variables and ai are constantcoefficients such that Linear processes have a wide range of applications in time series analysis.A large class of time series processes can be modelled in such a way,includinga subset of the fractional ARIMA processes (see Brockwell and Davis(1987).Additionally,we say that(Xn)is a short-range dependent process if the followingcondition is satisfied: It is obvious that ifand E(Z)<∞,then the linearprocess(Xn)is short-range dependent. K.Furmańczyk Let G be a measurable function such that E|G(X₁)|<∞.In statisticalinference for time series,it is an important problem to investigate the asymp-totic behavior of the partial sums,since many statis-tics are of this form.This problem has been considered by many authors.Forexample,Davydov(1970)considers the special case G(x)=x,which is alsodiscussed in Giraitis and Surgailis (1986).This problem has been consideredrecently by Ho and Hsing (1996),(1997),Hsing (2000),Koul and Surgailis(2001),Wu (2002). The central limit theorem for multiple linear processes with heavy tailedinnovations has been proved by Wu(2003).A very simple question is whetherone can prove a central limit theorem (CLT)under a short-range dependencewithout additional assumptions on the (a,)and (Zn)and for the most generalclass of functions G. The generalization of the one-sided linear process is the multidimensionallinear process(MLP),defined as follows: are i.i.d.random vectors in Rmwhere the innovationswith mean zero and E|Z₁I²=1. (1.1) ,Ao=I,and theThe(A)are the nonrandom matrices,wherematrix norm ||ll is such that,for every z∈R", (1.2) |A,z|≤||AIIIz|, where ||is the usual Euclidean norm.We will also consider the following assump-tions: (ao) for some t>1, (a₂) (b₁)the density of the vector Z₁is the Lipschitz function, (b₂(t))E|Z₁I<∞ for some t≥2. In Section 4 we prove the CLT for the sums of functionals of linearprocesses.Let ⑨[0,1]be the space of all real functions defined on [0,1]thatare right-continuous and have left-side limits with Skorokhod topology (seeBillingsley(1968)).We will show that,under some simple conditions, n-1/2Sm→σ{B(t),0≤t≤1} in ⑨[0,1]for some σ. The basic idea of this proof is the martingale decomposition of the process Yn=G(Xn)-EG(Xn). This decomposition is of the following form (see also Ho and Hsing(1996)and (1997)): where Un,s=E{Yn|n-s-E{Yn|9n-s-1},(1.3) where n:=σ(…,Zn-1,Zn)is the σ-field generated by the innovations in the“past”≤n,(Xn)is an m-dimensional process(MLP),and G:Rm→R is a realfunction such that E|Y₁I⁰<∞for some Q≥2. In the proof of the main theorem we use the idea introduced by Koul andSurgailis(1997). Theorem 2.2 is a simple corollary to the theorem of Maxwell and Wood-roofe (see Wu(2002),Corollary 1).In this context see also Ho and Hsing(1997).They proved that ifand(ao)holds,then under someregularity conditions on G (for example,if the derivative of G is bounded andcontinuous,or G is any polynomial) Sn/√n→N(0,o²) for some σ<0 (see Theorem 4.1 in Ho and Hsing (1997). On the other hand,under the assumptions that is Lipschitz for all sufficiently large n,an=n⁻L(n)for some y>1,where L()is a slowly varying function,Wu(2002)proved the CLT. ## 2.MAIN RESULTS The main results are the central limit theorems fo

FAQs

What type of time series models are considered in this paper?

The paper considers one-sided m-dimensional linear processes, and also introduces a related multidimensional linear process for the innovations vector.

What is the main probabilistic limit proved for functionals of the linear processes?

Under short-range dependence and Lipschitz-type regularity conditions on G, the normalized partial-sum process converges in distribution to σ times a standard Brownian motion on [0,1].

How does the proof establish the central limit theorem?

It uses a martingale decomposition of Yn=G(Xn)-E[G(Xn)], then combines martingale difference CLT results with weak convergence lemmas in the Skorokhod space.

Sep 15, 2026

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