Positive dependence in Markov chains (Q1063328)

Two random variables X and Y, with the same marginal distributions, are said to be positively dependent if cov(h(X),h(Y))\(\geq 0\) for every real function h such that \(E| h(x)h(y)| <\infty\). When X and Y are exchangeable then this notion reduces to the notion of positive definite dependence of the reviewer [Ann. Inst. Stat. Math. 31, 67-84 (1979; Zbl 0446.60011)]. Let \(\{X_ t\}=\{X_ t,t=0,1,2,...\}\) be a stationary ergodic irreducible aperiodic finite-state Markov chain with transition probability matrix T. The process \(\{X_ t\}\) is said to be positive dependent if \(X_ s\) and \(X_ t\) are positive dependent for all s and t. For such reversible Markov chains, the authors find equivalent conditions for the positive dependence of \(\{X_ t\}\). In particular they show that \(\{X_ t\}\) is positive dependent if all the characteristic roots of T are nonnegative. For such nonreversible Markov chains, the authors obtain necessary conditions for the positive dependence of \(\{X_ t\}.\) Finally the authors consider positive dependence of \(\{Y_ t,Z_ t,t=0,1,2,...\}\) where \(\{Y_ t,t=0,1,2,...\}\) and \(\{Z_ t,t=0,1,2,...\}\) are independent Markov chains. For example, they prove that if either \(\{Y_ t\}\) or \(\{Z_ t\}\) is reversible then \(\{(Y_ t,Z_ t)\}\) is positive dependent if and only if both \(\{Y_ t\}\) and \(\{Z_ t\}\) are positive dependent.