Convergence Rates for a Class of Large Deviation Probabilities

Abstract

For a sequence $\{X_n: 1 \leqq n < \infty\}$ of independent, identically distributed random variables with moment-generating functions, a 1952 theorem of Chernoff asserts that $n^{-1} \log P(S_n \geqq \lambda n) \rightarrow \log \rho$, where $S_n$ is the $n$th partial sum of the $X_k's \lambda > 0$, and $\rho$ is a constant depending on $\lambda$ and the distribution of $X_1$. A 1969 theorem of Sievers, as strengthened by Plachky in 1971, established the convergence of $n^{-1}\log P(W_n \geqq z_n)$ to a constant, where the $W_n$'s have moment-generating functions and belong to a class of random variables more general than partial sums, and the $z_n$'s are numbers such that $n^{-1}z_n \rightarrow \lambda > 0$. In a format related to that of Sievers, Behadur in 1971 analyzed the behavior of $n^{-1}\log P(W_n \geqq z_n)$ in situations when it may not converge to a constant. The goal of the present article is to extend the theorems of Chernoff, Sievers, and Bahadur in the direction of obtaining convergence rates (to 0) of the large deviation probabilities $P(W_n \geqq z_n)$ where the $z_n$'s are numbers such that $n^{-\frac{1}{2}} z_n \rightarrow \infty$. The method of proof is based on the proof of Chernoff's theorem given, in passing, in a 1960 paper of Bahadur and Ranga Rao.