## The representation of composition structures.(English)Zbl 0895.60037

Summary: A composition structure is a sequence of consistent probability distributions for compositions (ordered partitions) of $$n=1,2, \dots$$. Any composition structure can be associated with an exchangeable random composition of the set of natural numbers. Following P. Donnelly and P. Joyce [Adv. Appl. Probab. 23, No. 2, 229-258 (1991; Zbl 0724.60040)], we study the problem of characterizing a generic composition structure as a convex mixture of the “extreme” ones. We topologize the family $${\mathcal U}$$ of open subsets of $$[0,1]$$ so that $${\mathcal U}$$ becomes compact and show that $${\mathcal U}$$ is homeomorphic to the set of extreme composition structures. The general composition structure is related to a random element of $${\mathcal U}$$ via a construction introduced by J. Pitman [Probab. Theory Relat. Fields 102, No. 2, 145-158 (1995; Zbl 0821.60047)].

### MSC:

 60G09 Exchangeability for stochastic processes 60C05 Combinatorial probability 60J50 Boundary theory for Markov processes

### Citations:

Zbl 0724.60040; Zbl 0821.60047
Full Text:

### References:

 [1] Aldous, D. J. (1985). Exchangeability and related topics. Lecture Notes in Math. 1117 1- 198. Springer, Berlin. · Zbl 0562.60042 [2] Andrews, G. E. (1976). The Theory of Partitions. Addison-Wesley, Reading, MA. · Zbl 0655.10001 [3] Baryshnikov, Y., Eisenberg, B. and Stengle, G. (1995). A necessary and sufficient condition for the existence of the limiting probability of a tie for the first place. Statist. Probab. Lett. 23 211-220. · Zbl 0823.60029 [4] Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. in Appl. Probab. 23 229-258. JSTOR: · Zbl 0724.60040 [5] Kerov, S. V. (1989). Combinatorial examples in the theory of AF-algebras. In Zapiski Nauchnych Seminarov LOMI 172 55-67. Nauka, Leningrad. (In Russian.) · Zbl 0779.46046 [6] Kerov, S. V. (1995). Subordinators and permutation actions with a quasi-invariant measure. In Zapiski Nauchnych Seminarov POMI 223 181-218. Nauka, Sankt Petersburg. (In Russian.) · Zbl 0888.60015 [7] Kerov, S. V. and Vershik, A. M. (1981). Asymptotic theory of characters of the symmetric group. Functional Anal. Appl. 15 246-255. · Zbl 0507.20006 [8] Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. 18 374-380. · Zbl 0415.92009 [9] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248. · Zbl 0491.60076 [10] Pitman, J. (1995). Partition structures derived from Brownian motion and stable subordinators. Bernoulli. · Zbl 0882.60081 [11] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. · Zbl 0821.60047 [12] Vershik, A. (1995). Asymptotic combinatorics and algebraic analysis. In Proceedings of the International Congress on Mathematics, 1994, August 3-11, Z ürich 1384-1394. Birkhäuser, Basel. · Zbl 0843.05003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.