Higher order asymptotics for the Central Limit Theorem and Large Deviation Principles

Abstract

First, we present results that extend the classical theory of Edgeworth expansions to independent identically distributed non-lattice discrete random variables. We consider sums of independent identically distributed random variables whose distributions have (d+1) atoms and show that such distributions never admit an Edgeworth expansion of order d but for almost all parameters the Edgeworth expansion of order (d-1) is valid and the error of the order (d-1) Edgeworth expansion is typically O(n^{-d/2}) but the O(n^{-d/2}) terms have wild oscillations. Next, going a step further, we introduce a general theory of Edgeworth expansions for weakly dependent random variables. This gives us higher order asymptotics for the Central Limit Theorem for strongly ergodic Markov chains and for piece-wise expanding maps. In addition, alternative versions of asymptotic expansions are introduced in order to estimate errors when the classical expansions fail to hold. As applications, we obtain Local Limit Theorems and a Moderate Deviation Principle. Finally, we introduce asymptotic expansions for large deviations. For sufficiently regular weakly dependent random variables, we obtain higher order asymptotics (similar to Edgeworth Expansions) for Large Deviation Principles. In particular, we obtain asymptotic expansions for Cramer's classical Large Deviation Principle for independent identically distributed random variables, and for the Large Deviation Principle for strongly ergodic Markov chains.