A Note on First Exit Times with Applications to Sequential Analysis

Abstract

In this paper, we prove certain theorems about the first exit time $N=inf{n\geqq1:S_nT_n+R_n\not\in(-a,b)}$, where $S_n$ is the partial sum of i.i.d. random variables with zero mean and finite positive variance, and $R_n,T_n$ are two sequences of random variables satisfying certain conditions. Such exit times arise in the analysis of the stopping rules of invariant sequential probability ratio tests, and our theorems are then applied to study the stopping rules of these tests.