A Nonlinear Renewal Theory with Applications to Sequential Analysis I

Abstract

Renewal theory is developed for processes of the form $Z_n = S_n + \xi_n$, 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 $\mu$ and $\xi_n$ is independent of $X_{n+1}, X_{n+2}, \cdots$ and has sample paths which are slowly changing in an appropriate sense. Applications to sequential analysis are given.