Moderate Deviations of Hitting Times for Trajectories of Sums of Independent and Identically Distributed Random Variables

Abstract

In this paper we establish a moderate deviation principle of the hitting times for trajectories of sums of independent and identically distributed random variables. The main idea of proof is to convert the moderate deviations over a small time interval into the moderate deviations at a point, then utilize the moderate deviations for trajectories of sums of independent random variables given by Hu in \cite{Hu2001} to get the moderate deviations at a point. By the upper bounds of large deviations for trajectories of sums of independent random variables given by Schuette in \cite{ Schuette1994}, we can prove that our convert doesn't influence the rate function of the moderate deviation.