Conditional limit theorems under Markov conditioning

Ahtract -Let Xi, X, , . be independent identically distributed random variables taking values in a finite set X and consider the conditional joint distribution of the first m elements of the sample Xt; . ., X, on the condition that A’, = x, and the sliding block sample average of a function h( ., .) defined on X2 exceeds a threshold OL > Eh( Xt, X2). For m fixed and M + co, this conditional joint distribution is shown to converge to the m-step joint distribution of a Markov chain started in x1 which is closest to X,, X2, in Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distribution P( , .) satisfies EP( x, y ) h( x, y) 2 OL, provided some distribution P on X2 having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when X,, X2, . . . is an arbitrary finite-order Markov chain and more general conditioning is allowed.