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

Feller coupling of cycles and Poisson spacings