Conditional limit theorems under Markov conditioning
- 1 November 1987
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 33 (6) , 788-801
- https://doi.org/10.1109/tit.1987.1057385
Abstract
LetX_{1},X_{2},cdotsbe independent identically distributed random variables taking values in a finite setXand consider the conditional joint distribution of the first m elements of the sampleX_{1},cdots , X_{n}on the condition thatX_{1}=x_{1}and the sliding block sample average of a functionh(cdot , cdot)defined onX^{2}exceeds a thresholdalpha > Eh(X_{1}, X_{2}). Formfixed andn rightarrow infty, this conditional joint distribution is shown to converge m them-step joint distribution of a Markov chain started inx_{1}which is closest toX_{l}, X_{2}, cdotsin Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distributionP(cdot , cdot)satisfiessum P(x, y)h(x, y)geq alpha, provided some distributionPonX_{2}having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained whenX_{1}, X_{2},cdotsis an arbitrary finite-order Markov chain and more general conditioning is allowed.Keywords
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