Central Limit Theorems for dependent variables. I

Abstract

This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak 'mixing' condition called f-mixing.Roughly speaking, an array of real r.v.s is said to be f-mixing if linear combinations of its 'past' and 'future' are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, p-mixing and 0-mixing) are special cases of f-mixing. Linear processes are shown to be f-mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are 'nearly second order stationary'.

Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables

“…On the other hand, not all weakly dependent sequences and triangular arrays that are known to satisfy LLNs and the central limit theorem are mixing. For example, linear processes including fi rst-order autoregressive processes are not necessarily <!>O. p(·), or o.O mixing, see Andrews (1984Andrews ( , 1985 (although many such processes are aO mixing, see Withers (1981) and Goredetskii (1977)). integrable L 1-mixingale with ci = suv II E.1; Iii for i :2'.…”

Central limit theorem for sampled sums of dependent random variables

“…They assume that {|ξ i | 2+δ } is uniformly integrable for a certain δ > 0. Such an assumption is also required for Theorem 2.1 in [27] for l-mixing arrays. In [5], the random variables ξ i are assumed to be uniformly bounded.…”

Functional least squares estimators in an additive effects outliers model

“…We shall now show that the sup norm of the second term in (3.11) is o p {\). It can be dominated by Sunil K. Dhar [12] the third and fourth terms in (3.16) goes to 0 in probability. Since the sine and cosine functions satisfy similar properties, to prove that the sup norm of the expression in (3.16) converges to zero in probability we only need now prove that the sup norm of the second term in (3.16) converges to zero in probability.…”

“…The C.L.T. given by [12] and [13] has been used to prove its finite dimensional distribution convergence. We also discuss the behavior of its asymptotic bias.…”