Identifying a large deviation rate function. (English) Zbl 0777.60024

Assume that a sequence \(\{P_ n\}_{n\in\mathbb{N}}\) of probability measures satisfies a large deviation principle with rate function \(I\). It is proved that \(I\) is unique and that \(I\) is given by a variational expression analogous to the one defining the convex conjugate. Two examples of Markov chains with non-convex rate function are given. If \(I\) is also assumed to be convex, then \(I\) is indeed the convex conjugate of an explicitly defined function. These results are applied to the empirical laws of a Markov chain; together with results of de Acosta and Nummelin they yield universal upper and lower bounds on \(I\). Two examples of Markov chains are given in which the empirical laws have a large deviation rate function coinciding either with the lower or the upper bound, or lying strictly between these two, depending on the starting point. Finally, a criterion is given for the rate function to coincide with the upper bound.


60F10 Large deviations
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