Translator Disclaimer
2020 Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials
Joseph Najnudel, Jim Pitman
Electron. Commun. Probab. 25: 1-11 (2020). DOI: 10.1214/20-ECP352

Abstract

Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller’s coupling, to associate cycles of random permutations governed by the Ewens $(\theta )$ distribution with spacings derived from independent Bernoulli trials with success probability $\theta /(n-1+\theta )$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length $\ell $ are independent Poisson variables with means $\theta /\ell $. Ignatov (1978) first discovered this remarkable result in the uniform case $\theta = 1$, by constructing Bernoulli $(1/n)$ trials as the indicators of record values in a sequence of i.i.d. uniform $[0,1]$ variables. In the present article, the Poisson property of inhomogeneous Bernoulli spacings is explained by a variation of Ignatov’s approach for a general $\theta >0$. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

Citation

Download Citation

Joseph Najnudel. Jim Pitman. "Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials." Electron. Commun. Probab. 25 1 - 11, 2020. https://doi.org/10.1214/20-ECP352

Information

Received: 23 July 2019; Accepted: 22 September 2020; Published: 2020
First available in Project Euclid: 2 October 2020

MathSciNet: MR4158233
Digital Object Identifier: 10.1214/20-ECP352

Subjects:
Primary: 60C05

JOURNAL ARTICLE
11 PAGES


SHARE
Back to Top