Strong Large Deviation and Local Limit Theorems

Abstract : An event is loosely called a large deviation event if the dominant term in the probability of that event goes to zero exponentially. Most papers give asymptotic expressions only to the algorithm of the probability of a large deviation event. This paper obtains strong deviation results which give asymptotic expressions to the actual probability of a large deviation event based on an arbitrary sequence of random variables, under some conditions on the moment generating functions. The proof of these results depends on local limit theorems, which also are proved in this paper, by imposing some conditions on the characteristic functions. A local limit theorem states that the pseudo- densities of random variables converge, which is stronger than the convergence of these random variables in distribution.