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June, 1977 Spacing Distribution Associated with a Stationary Random Measure on the Real Line
Sidney C. Port, Charles J. Stone
Ann. Probab. 5(3): 387-394 (June, 1977). DOI: 10.1214/aop/1176995799


Let $\mathcal{N}$ denote the collection of all Radon measures $n$ on $\mathbb{R}$ such that $0 < \lim_{x\rightarrow\infty} n((0, x\rbrack)/x = \lim_{x\rightarrow -\infty}n((x, 0\rbrack)/|x| < \infty$. For $n\in\mathscr{N}$, let $n^{-1}\in\mathscr{N}$ be the measure whose distribution function is the inverse of the distribution function of $n$. Given a random element $N$ of $\mathscr{N}$ having distribution $P$, let $P^I$ denote the distribution of $N^{-1}$. Let $N$ be a random element of $\mathscr{N}$ having stationary distribution $P$ and let $P^T$ be the appropriately defined tagged distribution corresponding to $P$. It is shown that $P^I$ has an asymptotically stationary distribution $P^S$ on $\mathscr{N}$. Moreover $P = (P^S)^S, P^I = (P^S)^T$, and $P^T = (P^S)^I. P^S$ is given explicitly in terms of $P^T$. In particular, if $N$ is purely nonatomic with probability one, then $P^S = (P^T)^I$. If $P$ is a stationary compound renewal process, then so is $P^S$.


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Sidney C. Port. Charles J. Stone. "Spacing Distribution Associated with a Stationary Random Measure on the Real Line." Ann. Probab. 5 (3) 387 - 394, June, 1977.


Published: June, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0366.60084
MathSciNet: MR443079
Digital Object Identifier: 10.1214/aop/1176995799

Primary: 60K99
Secondary: 60K05

Keywords: compound renewal process , Palm measure , Poisson process , random measure , spacing distribution , tagged distribution

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 3 • June, 1977
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