Interacting reinforced-urn systems

We introduce a class of discrete-time stochastic processes generated by interacting systems of reinforced urns. We show that such processes are asymptotically partially exchangeable and we prove a strong law of large numbers. Examples and the analysis of particular cases show that interacting reinforced-urn systems are very flexible representations for modelling countable collections of dependent and asymptotically exchangeable sequences of random variables.

[1]  D. Aldous.  Exchangeability and related topics , 1985 .

[2]  Piercesare Secchi,et al.  Urn schemes and reinforced random walks , 2000 .

[3]  B. M. Hill,et al.  Exchangeable Urn Processes , 1987 .

[4]  Stephen G. Walker,et al.  A bivariate Dirichlet process , 2003 .

[5]  G. Pólya,et al.  Über die Statistik verketteter Vorgänge , 1923 .

[6]  D. Freedman,et al.  De Finetti's Theorem for Markov Chains , 1980 .

[7]  William D. Sudderth,et al.  A Strong Law for Some Generalized Urn Processes , 1980 .

[8]  École d'été de probabilités de Saint-Flour,et al.  École d'été de probabilités de Saint-Flour XIII - 1983 , 1985 .

[9]  Shi Wujie,et al.  A characteristic property of PSL2(7) , 1984, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[10]  Anna Maria Paganoni,et al.  On a two-color generalized Polya urn , 2005 .

[11]  B. Friedman,et al.  A simple urn model , 1949 .

[12]  K. Doksum.  Tailfree and Neutral Random Probabilities and Their Posterior Distributions , 1974 .

[13]  Partially exchangeable processes indexed by the vertices of a k-tree constructed via reinforcement , 2005 .

[14]  R. Pemantle,et al.  A time-dependent version of Pólya's urn , 1990 .

[15]  Stephen G. Walker,et al.  Reinforced random processes in continuous time , 2003 .

[16]  S. Walker,et al.  Beta-Stacy processes and a generalization of the Polya urn scheme , 1997 .

[17]  J. Neveu,et al.  Martingales à temps discret , 1973 .