Limit theorems for random triangular urn schemes
Aguech Rafik
Source: J. Appl. Probab. Volume 46, Number 3 (2009), 827-843.
Abstract
In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.
Primary Subjects: 60J80, 60J85, 60G40, 62G20
Secondary Subjects: 60K05, 60G46, 60F05
Keywords: Multitype branching process; generalized Pólya urn; urn model; Yule process
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1253279854
Digital Object Identifier: doi:10.1239/jap/1253279854
Zentralblatt MATH identifier:
05611420
References
Aldous, D., Flannery, B. and Palacios, J. L. (1988). Two applications of urn processes: the fringe analysis of search trees and the simulation of quasi-stationary distributions of Markov chains. Prob. Eng. Inf. Sci. 2, 293--307.
Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching process and related limit theorems. Ann. Math. Statist. 39, 1801--1817.
Mathematical Reviews (MathSciNet):
MR232455
Digital Object Identifier: doi:10.1214/aoms/1177698013
Project Euclid: euclid.aoms/1177698013
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
Mathematical Reviews (MathSciNet):
MR373040
Bagchi, A. and Pal, A. K. (1985). Asymptotic normality in the generalized Pólya--Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Meth. 6, 394--405.
Mathematical Reviews (MathSciNet):
MR791169
Digital Object Identifier: doi:10.1137/0606041
Bernstein, S. (1940). Sur un probléme du schéma des urnes à composition variable. C. R. (Doklady) Acad. Sci. URSS 28, 5--7.
Mathematical Reviews (MathSciNet):
MR3507
Eggenberger, F. and Pòlya, G. (1923). Über die statistik verketetter vorgäge. Z. Angew. Math. Mech. 1, 279--289.
Flajolet, P. and Vincent, P. (2005). Triangular urns with dimension 2 and 3. Doctoral Thesis, École Polytechnique.
Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Prob. 33, 1200--1233.
Mathematical Reviews (MathSciNet):
MR2135318
Zentralblatt MATH:
1073.60007
Digital Object Identifier: doi:10.1214/009117905000000026
Project Euclid: euclid.aop/1115386724
Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Prob. 21, 1624--1639.
Mathematical Reviews (MathSciNet):
MR1235432
Digital Object Identifier: doi:10.1214/aop/1176989134
Project Euclid: euclid.aop/1176989134
Harris, T. E. (1951). Some mathematical models for branching processes. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, pp. 305--328.
Mathematical Reviews (MathSciNet):
MR45331
Zentralblatt MATH:
0045.07701
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177--245.
Mathematical Reviews (MathSciNet):
MR2040966
Digital Object Identifier: doi:10.1016/j.spa.2003.12.002
Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417--452.
Mathematical Reviews (MathSciNet):
MR2226887
Zentralblatt MATH:
1112.60012
Digital Object Identifier: doi:10.1007/s00440-005-0442-7
Karlin, S. and McGregor, J. (1968). Embeddability of discrete time simple branching processes into continuous time branching processes. Trans. Amer. Math. Soc. 132, 115--136.
Mathematical Reviews (MathSciNet):
MR222966
Zentralblatt MATH:
0159.46102
Digital Object Identifier: doi:10.2307/1994885
Kotz, S., Mahmoud, H. and Robert, P. (2000). On generalized Pólya urn models. Statist. Prob. Lett. 49, 163--173.
Mathematical Reviews (MathSciNet):
MR1790166
Pólya, G. (1931). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1, 117--161.
Mathematical Reviews (MathSciNet):
MR1507985
Savkevitch, V. (1940). Sur le schéma des urnes à composition variable. C. R (Doklady) Acad. Sci. URSS 28, 8--12.
Mathematical Reviews (MathSciNet):
MR3506
Journal of Applied Probability
