J. Appl. Prob. 46, 827–843 (2009) Printed in England  Applied Probability Trust 2009 LIMIT THEOREMS FOR RANDOM TRIANGULAR URN SCHEMES RAFIK AGUECH, ∗ Faculté des Sciences de Monastir,Tunisia 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. Keywords: Multitype branching process; generalized Pólya urn; urn model; Yule process 2000 Mathematics Subject Classification: Primary 60J80; 60J85; 60G40; 62G20 Secondary 60K05; 60G46; 60F05 1. Introduction We consider a generalized Pólya urn with balls of two colors, say white (W) and black (B). The urn is initially nonempty. At each time n, a ball is drawn from the urn uniformly at random and its color is observed (i.e. a ball is drawn, looked at, and then placed back into the urn). If a white ball is drawn then it is replaced in the urn with X n white balls; if a black ball is drawn then it is replaced in the urn with Y n white balls and Z n black balls. The random variables X n , Y n , and Z n are independent copies of some nonnegative, integer-valued random variables X, Y , and Z, respectively. The evolution rule at time n is then summarized by the 2 × 2 random matrix  W B W X n 0 B Y n Z n  , where the rows indicate the number of balls added to the urn and the columns indicate the number of balls drawn. Thus, the composition of the urn after n draws is represented by the vector (W n ,B n ), where W n and B n are the numbers of white and black balls, respectively, in the urn. The urn starts with a given vector (W 0 ,B 0 ), which we assume is nonrandom. The assumption that X, Y , and Z are nonnegative, integer-valued random variables guaran- tees the nonextinction of the urn. Furthermore, in order to avoid any explosion of the urn, we suppose that X, Y , and Z have finite variances. We define μ X = E(X), μ Y = E(Y ), μ Z = E(Z), σ 2 X = var(X), σ 2 Y = var(Y ), σ 2 Z = var(Z), Received 27 June 2007; revision received 6 July 2009. ∗ Postal address: Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir et EPAM Sousse, Tunisia. Email address: rafik.aguech@ipeit.rnu.tn 827 available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0021900200005908 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 02:01:04, subject to the Cambridge Core terms of use,