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Regenerative composition structures. (English) Zbl 1070.60034

A composition of \(n\) with parts \(n_1,\dots, n_k\) is an ordered sequence \((n_1,\dots, n_k)\) of positive integers with sum \(n\). A random composition is a random variable \(C_n\) with values in the set of compositions of \(n\). A composition structure \((C_n)\) is a Markovian sequence \(C_n\), \(n= 1,2,\dots\), of random compositions having sampling consistency: \(n\) identical balls are distributed into an ordered series of boxes according to \(C_n\) and \(C_{n-1}\) is obtained by discarding one ball at random and deleting any empty box created. The composition structure \((C_n)\) is regenerative if, given that the first part \(F_n\) of \(C_n\) is \(m\), the remaining composition of \(n-m\) is distributed as \(C_{n-m}\), \(1\leq m< n\). Then \(P(C_n= (n_1,\dots, n_k))\) is a product of decrement matrix elements \(q(h; m)= P(F_h =m)\). This is also sufficient for \((C_n)\) to be regenerative. The \(q(n; m)\) then satisfy a recurrence.
In the above boxes label the balls at random from 1 to \(n\). This gives a random ordered set-partition \(C_n^*\) of \(\{1,\dots,n\}\). The construction of the \(C^*_n\) is consistent by the sampling consistency of \((C_n)\) and so defines a random ordered partition \(C^*\) of \(N\). The regenerative composition structures are characterized (Theorem 5.2) as follows: Let \(R\) be the closed range of a subordinator \(S_t\), \(t\geq 0\). Let \(\varepsilon_1, \varepsilon_2,\dots\) be i.i.d. standard exponentials independent of the \(S_t\) and \(\varepsilon_{1n},\dots, \varepsilon_{nn}\) the increasing order statistics of \(\varepsilon_1,\dots,\varepsilon_n\). The \(j\in \{1,\dots, n\}\) with \(\varepsilon_{jn}\) in a same component of \(R^c\) then define a block of an ordered set-partition of \(\{1,\dots, n\}\), each \(i\) with \(\varepsilon_{in}\in R\) being a singleton of this partition. The sizes of the blocks form a regenerative composition of \(n\).
Further subjects studied are: Examples, e.g. \(X_{hk}\) i.i.d. 0-1 variables. The \(i\)th block is the set of \(h\) with epoch \(i\) of first 0. Or points of an i.i.d. sequence falling into random intervals of \([0,1]\). A two-parameter family of composition structures starting from \(C^*_n\). The effect of the transformation \(z\to 1-\exp(-z)\) giving the random set \(\widetilde R= 1-\exp(-R)\). The \(q(n;n)\) determine the distribution of regenerative \((C_n)\). The frequency of singletons in \(C^*\). Symmetry between \((n_1,\dots, n_k)\) and \((n_k,\dots, n_1)\) or, equivalently, \(\widetilde R\) and \(1-\widetilde R\). A closer study of the intervals of \(R^c\) and sample points \(\varepsilon_{jn}\) inducing a partition of \([0,\infty)\).

MSC:

60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
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