Cramér asymptotics in a system with slow and fast Markovian motions

A cascade with fast and slow motions is considered in which the rapid motions constitute an ergodic Markov chain. Asymptotics of the probabilities of large deviations of the Cramér type are calculated for the difference between the slow component of the cascade trajectory and some averaged trajectory. The Taylor expansions in the powers of a small parameter are calculated for the semi-invariants of the difference which are smoothly dependent on time.