Binary sequential representations of random partitions
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.
scientific article; zbMATH DE number 5014665
scientific article