Strong Large Deviation and Local Limit Theorems

Abstract

Most large deviation results give asymptotic expressions for $\log P(Y_n \geq y_n)$, where the event $\{Y_n \geq y_n\}$ is a large deviation event, that is, $P(Y_n \geq y_n)$ goes to 0 exponentially fast. We refer to such results as weak large deviation results. In this paper we obtain strong large deviation results for arbitrary random variables $\{Y_n\}$, that is, we obtain asymptotic expressions for $P(Y_n \geq y_n)$, where $\{Y_n \geq y_n\}$ is a large deviation event. These strong large deviation results are obtained for lattice valued and nonlattice valued random variables and require some conditions on their moment generating functions. These results strengthen existing results which apply mainly to sums of independent and identically distributed random variables. Since $Y_n$ may not possess a probability density function, we consider the function $q_n(y; b_n,S) = \lbrack(b_n/\mu(S))P(b_n(Y_n - y) \in S)\rbrack$, where $b_n \rightarrow \infty, \mu$ is the Lebesgue measure on $R$, and $S$ is a measurable subset of $R$ such that $0 < \mu(S) < \infty$. The function $q_n(y; b_n,S)$ is the p.d.f. of $Y_n + Z_n$, where $Z_n$ is uniform on $-S/b_n$, and will be called the pseudodensity function of $Y_n$. By a local limit theorem we mean the convergence of $q_n(y_n; b_n,S)$ as $n \rightarrow \infty$ and $y_n \rightarrow y^\ast$. In this paper we obtain local limit theorems for arbitrary random variables based on easily verifiable conditions on their characteristic functions. These local limit theorems play a major role in the proofs of the strong large deviation results of this paper. We illustrate these results with two typical applications.