Feller coupling of cycles and Poisson spacings

arXiv1907.09587MaRDI QIDQ6322521

Jim Pitman, Joseph Najnudel

Publication date: 22 July 2019

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 $(heta)$ distribution with spacings derived from independent Bernoulli trials with success probability $heta/(n-1+heta)$ 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 $heta/ell$. Ignatov (1978) first discovered this remarkable result in the uniform case $heta=1$, 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 $heta>0$. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

Mathematics Subject Classification ID

Combinatorial probability (60C05)

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