Conditional limit theorems under Markov conditioning

Let X_{1},X_{2},\cdots 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 X_{1},\cdots , X_{n} on the condition that X_{1}=x_{1} and the sliding block sample average of a function h(\cdot , \cdot) defined on X^{2} exceeds a threshold \alpha > Eh(X_{1}, X_{2}) . For m fixed and n \rightarrow \infty , this conditional joint distribution is shown to converge m the m -step joint distribution of a Markov chain started in x_{1} which is closest to X_{l}, X_{2}, \cdots in Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distribution P(\cdot , \cdot) satisfies \sum P(x, y)h(x, y)\geq \alpha , provided some distribution P on X_{2} having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when X_{1}, X_{2},\cdots is an arbitrary finite-order Markov chain and more general conditioning is allowed.

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