Nonlinear Markov Renewal Theory with Statistical Applications

Abstract

An analogue of the Lai-Siegmund nonlinear renewal theorem is proved for processes of the form $S_n + \xi_n$, where $\{S_n\}$ is a Markov random walk. Specifically, $Y_0,Y_1,\cdots$ is a Markov chain with complete separable metric state space; $X_1,X_2,\cdots$ is a sequence of random variables such that the distribution of $X_i$ given $\{Y_j, j \geq 0\}$ and $\{X_j, j \neq i\}$ depends only on $Y_{i - 1}$ and $Y_i; S_n = X_1 + \cdots + X_n$; and $\{\xi_n\}$ is slowly changing, in a sense to be made precise below. Applications to sequential analysis are given with both countable and uncountable state space.