Conditional limit theorems under Markov conditioning

Let be independent identically distributed random variables taking values in a finite set and consider the conditional joint distribution of the first m elements of the sample on the condition that and the sliding block sample average of a function defined on exceeds a threshold . For fixed and , this conditional joint distribution is shown to converge m the -step joint distribution of a Markov chain started in which is closest to in Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distribution satisfies , provided some distribution on having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when is an arbitrary finite-order Markov chain and more general conditioning is allowed.