The Large Deviations of Estimating Rate-Functions

Given a sequence of bounded random variables that satisfies a well known mixing condition, it is shown that empirical estimates of the ratefunction for the partial sums process satisfies the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue-length distribution at a single server queue with infinite waiting space is proved.