循环的Feller耦合与Poisson间距
Feller coupling of cycles and Poisson spacings
摘要：
Feller（1945）给出了在$[n]$的均匀随机排列中不同大小的循环的计数与在一系列$n$独立伯努利试验中成功间隔之间的耦合，其中在$n$第次试验中成功概率为$1/n$。Arratia、Barthel和Tavaré（1992）扩展了Feller耦合，将由Ewens $(θ)$分布控制的随机排列的循环与从独立伯努利试验中导出的间隔相关联，在$n$第次试验中成功概率为$θ/(n-1+θ)$，并得出结论，在这样的试验的无限序列中，长度为$\ell$的间隔的数量是具有平均值$θ/\ell$的独立泊松变量。Ignatov（1978）首先在统一的$θ= 1$情况下发现了这个显著的结果，通过构造伯努利$(1/n)$试验作为i.i.d.序列中记录值的指标。均匀$[0,1]$变量。在本文中，泊松性质的非齐次伯努利空间的解释的一个变化的Ignatov的方法为一般的$θ>0$。此外，我们的方法自然地提供了随机排列的无限集，其周期计数是完全由独立的泊松随机变量。
英文摘要：
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 $(θ)$ distribution with spacings derived from independent Bernoulli trials with success probability $θ/(n-1+θ)$ 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 $θ/\ell$. 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.
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