## 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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### References:

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