Spacing Distribution Associated with a Stationary Random Measure on the Real Line

Abstract

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 $P^S$.