On the Nonlinear Renewal Theorem

Abstract

Let $Z_1, Z_2,\ldots$ be jointly distributed random variables for which $\sup_k Z_k = \infty \mathrm{w.p.}1$ and let $t = t_a = \inf(n \geq 1: Z_n > a)$ and $R_a = Z_t - a$ for $a \geq 0$. Conditions under which $R_a$ has a limiting distribution as $a \rightarrow \infty$ are developed. These require that the finite dimensional, conditional distributions of the increments $Z_{t+k} - Z_t, k \geq 1$, converge to the finite dimensional distributions of a process for which the result is known, thus weakening the slow change condition in earlier work. The main result is applied to some sequences for which the limiting distributions are those of the partial sums of an exchangeable process. These include the Euclidean norms of a driftless random walk in several dimensions and sequences for which the conditional distribution of $Z_{n+1} - Z_n$ given the past has a limit $\mathrm{w.p.}1$ as $n \rightarrow \infty$.