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 l-mixing. Roughly speaking, an array of real r.v.s is said to be l-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, ϱ-mixing and ψ-mixing) are special cases of l-mixing. Linear processes are shown to be l-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’.