### A Nonlinear Renewal Theory with Applications to Sequential Analysis II

T. L. Lai and D. Siegmund
Source: Ann. Statist. Volume 7, Number 1 (1979), 60-76.

#### 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.

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Primary Subjects: 62L10
Secondary Subjects: 60K05
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