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October, 1990 On the Nonlinear Renewal Theorem
Michael Woodroofe
Ann. Probab. 18(4): 1790-1805 (October, 1990). DOI: 10.1214/aop/1176990649


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 $n \rightarrow \infty$.


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Michael Woodroofe. "On the Nonlinear Renewal Theorem." Ann. Probab. 18 (4) 1790 - 1805, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0717.60102
MathSciNet: MR1071826
Digital Object Identifier: 10.1214/aop/1176990649

Primary: 60K05

Keywords: excess over the boundary , exchangeable processes , first passage times , Limiting distribution , Random walks

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • October, 1990
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