Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials

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 nth trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens (θ) distribution with spacings derived from independent Bernoulli trials with success probability θ/(n−1+θ) at the nth trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length are independent Poisson variables with means θ/ . Ignatov (1978) first discovered this remarkable result in the uniform case θ = 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 θ > 0. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

Stochastic Couplings and Bijections from the Symmetric Group to Itself

“…In [6] they study the Feller Coupling for random derangement of [n] under the Ewens distribution with parameter θ arising as the weak limit as n → ∞. Then, in [10], they describe a nice construction of said permutation in the case that θ = 1, and finally, [12] extends this approach for general θ > 0.…”