Conditional limit theorems under Markov conditioning

Let X₁, X₂, 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 X₁, , X₍ on the condition that X₁=x₁ and the sliding block sample average of a function h (, ) defined on X^2 exceeds a threshold > Eh (X₁, X₂). For m fixed and n, this conditional joint distribution is shown to converge m 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 P (x, y) h (x, y), provided some distribution P on X₂ having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when X₁, X₂, is an arbitrary finite-order Markov chain and more general conditioning is allowed.