A Nonlinear Renewal Theory with Applications to Sequential Analysis II

Abstract

This paper continues earlier work of the authors. An analogue of Blackwell's renewal theorem is obtained for processes $Z_n = S_n + \xi_n$, where $S_n$ is the $n$th partial sum of a sequence $X_1, X_2, \cdots$ of independent identically distributed random variables with finite positive mean and $\xi_n$ is independent of $X_{n+1}, X_{n+2}, \cdots$ and has sample paths which are slowly changing in a sense made precise below. As a consequence, asymptotic expansions up to terms tending to 0 are obtained for the expected value of certain first passage times. Applications to sequential analysis are given.