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.
"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