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Binary sequential representations of random partitions. (English) Zbl 1093.60008

From the author’s text: “A partition of \(n\) is simply a collection of positive integers whose sum is \(n\). Let \(\omega_n\) denote the collection of partitions of \(n\) and encode a partition \(\pi\in\omega_n\) via the component counts \((a_1,\dots ,a_n)\), where \(a_i\) is the number of components of size \(i\), for \(i=1,\dots,n\). A partition structure is a sequence of probability measures \(P_1,P_2,\dots\) defined on \(\omega_1,\omega_2,\dots\), respectively, which satisfies the consistency relation \[ \begin{aligned} P_n(a_1,\dots,a_n) = &P_{n+1}(a_1+1,\dots,a_{n+1})\frac{a_1+1}{n+1}\\ &+\sum_{r=2}^{n+1} P_{n+1}(a_1,\dots ,a_{r-1}-1,a_r+1,\dots,a_{n+1}) \frac{r(a_r+1)}{n+1} \end{aligned} \] for all \(n\geq 1\). Partition structures are naturally constructed by sampling individuals of various types from hypothetical infinite populations. If we allow these populations to be random and have ‘novel’ types of individuals, then all partition structures may be constructed in this fashion.”
Let \(\xi=(\xi_1,\xi_2,\dots)\) be a sequence of independent Bernoulli random variables with success probability \(p_n=P(\xi_n=1)\). Define for each \(n\geq 1\) a partition of \([n]=\{1,\dots,n\}\) by \(\Pi_n(\xi)=(a_1,\dots,a_n)\in\omega_n\), where \(a_i\) is the number of \(i\)-spacings between successive \(1\)s in the finite sequence \((\xi_1,\dots,\xi_n)\). It turns out that the sequence \((P_n)\) of distributions of the partitions \(\Pi_n\) forms a random partition if and only if \(p_n=\theta/(\theta+n-1)\) for \(n\geq 1\) with \(\theta=p_2/(1-p_2)\). This is the Ewens’ one-parameter family of partitions. The author obtains a two-parameter generalization of Ewens’ partition by considering random partitions constructed from discrete renewal processes and introducing a convolution-type product on \(0-1\) sequences.

MSC:

60C05 Combinatorial probability
05A18 Partitions of sets
Full Text: DOI

References:

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