Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T13:55:15.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Interacting reinforced-urn systems

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Anna Maria Paganoni  and
Piercesare Secchi
Anna Maria Paganoni*
Affiliation:
Politecnico di Milano
Piercesare Secchi*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Matematica ‘F. Brioschi', Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20123 Milano, Italy.
∗ Postal address: Dipartimento di Matematica ‘F. Brioschi', Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20123 Milano, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Urn modelreinforcementPólya urn

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60G09: Exchangeability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 791 - 804
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. J. (1985). Exchangeability and related topics. In École d'Été de Probabilités de Saint-Flour, XIII—1983 (Lecture Notes Math. 1117), Springer, Berlin, pp. 1–198.Google Scholar
[2] Athreya, K. B. (1969). On a characteristic property of Pólya's urn. Studia Sci. Math. Hung. 4, 31–35.Google Scholar
[3] Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Prob. 2, 183–201.Google Scholar
[4] Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. Angewandte Math. Mech. 3, 279–289.Google Scholar
[5] Friedman, B. (1949). A simple urn model. Commun. Pure Appl. Math. 2, 59–70.Google Scholar
[6] Hill, B. M., Lane, D. and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8, 214–226.Google Scholar
[7] Hill, B. M., Lane, D. and Sudderth, W. (1987). Exchangeable urn processes. Ann. Prob. 15, 1586–1592.Google Scholar
[8] May, C., Paganoni, A. M. and Secchi, P. (2002). On a two-color generalized Pólya urn. Quaderno di Dipartimento 505/P, Dipartimento di Matematica, Politecnico di Milano.Google Scholar
[9] Muliere, P., Secchi, P. and Walker, S. G. (2000). Urn schemes and reinforced random walks. Stoch. Process. Appl. 88, 59–78.CrossRefGoogle Scholar
[10] Muliere, P., Secchi, P. and Walker, S. G. (2003). Partially exchangeable processes indexed by the vertices of a (k) tree constructed via reinforcement. Quaderno di Dipartimento 543/P, Dipartimento di Matematica, Politecnico di Milano.Google Scholar
[11] Muliere, P., Secchi, P. and Walker, S. G. (2003). Reinforced random processes in continuous time. Stoch. Process. Appl. 104, 117–130.CrossRefGoogle Scholar
[12] Neveu, J. (1972). Martingales à Temps Discret. Masson, Paris.Google Scholar
[13] Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.Google Scholar
[14] Walker, S. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya urn scheme. Ann. Statist. 25, 1762–1780.Google Scholar
[15] Walker, S. G. and Muliere, P. (2003). A bivariate Dirichlet process. Statist. Prob. Lett. 64, 1–7.Google Scholar