Asymptotic theory of large deviations for Markov chains

A formal asymptotic expansion is constructed for the joint probability density function (pdf) of a stationary ergodic Markov chain, X-n, and its averages, Y-n. Since the pair (X-n; Y-n) is Markovian, the joint pdf satisfies a forward Kolmogorov equation whose solution is expanded asymptotically for large n. An algorithm is proposed for the calculation of the full asymptotic series, but only the three leading terms are found explicitly. It is found that for small values of the average, the asymptotic expansion coincides with the appropriate version of the central limit theorem. The ideas and methods are generalized to a large class of averages and to vector valued Markov chains.