Limit Theorems for Associated Random Variables

Abstract

A sequence of partial sums of mean zero associated random variables forms a demimartingale. We have discussed properties of demimartingales in Chapter 2 and some probabilistic properties of associated sequences in Chapter 1. Let $\{\Omega, \mathcal{F}, \mathcal{P}\}$ be a probability space and {X _n ,n≥1} be a sequence of associated random vari- ables defined on it. Recall that a finite collection {X ₁,X ₂,…,X _n } is said to be associated if for every pair of functions h(x) and g(x) from R ⁿ to R, which are nondecreasing componentwise,

$$\mathrm{Cov}(h(\mathbf{X}), \mathbf{g}(\mathbf{X})) \geq \boldsymbol{0},$$

whenever it is finite, where X=(X ₁,X ₂,…,X _n ) and an infinite sequence {X _n ,n≥1} is said to be associated if every finite subfamily is associated. As we have mentioned in Chapter 1, associated random variables are of considerable interest in reliability studies (cf. Esary, Proschan and Walkup (1967), Barlow and Proschan (1975)), statistical physics (cf. Newman (1980, 1983)) and percolation theory (cf. Cox and Grimmet (1984)). We have given an extensive review of several probabilistic results for associated sequences in Chapter 1 (cf. Prakasa Rao and Dewan (2001), Roussas (1999)). We now discuss some recent advances in limit theorems for associated random variables. Covariance inequalities of different types play a major role in deriving limit theorems for partial sums of associated random variables. The next section gives some covariance inequalities and their applications.