Identifying a Large Deviation Rate Function

Abstract

Assume a sequence of probabilities $\{P_n\}$ has a large deviation rate function $I$. It is proved that $I$ takes a form analogous to a convex conjugate. If $I$ is also assumed convex, then $I$ is a convex conjugate of an explicitly defined function $\psi$. The results are applied to the empirical law of a Markov chain yielding universal bounds on $I$. Examples are given of Markov chains in which the empirical law has a large deviation rate strictly between the given bounds.