Feller coupling of cycles and Poisson spacings

TL;DR

This paper explains the Poisson distribution of cycle counts in random permutations through a variation of Ignatov's approach, extending Feller's coupling to inhomogeneous Bernoulli spacings for general parameters.

Contribution

It introduces a new explanation for the Poisson property of inhomogeneous Bernoulli spacings and constructs infinite permutations with cycle counts as independent Poisson variables.

Findings

Poisson distribution of cycle counts in permutations is explained via a new coupling.

Extension of Feller's coupling to inhomogeneous Bernoulli trials for general  > 0.

Construction of infinite permutations with cycle counts as independent Poisson variables.

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 Tavar\'e (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.