## Bernoulli

- Bernoulli
- Volume 15, Number 1 (2009), 279-295.

### Multicolor urn models with reducible replacement matrices

Arup Bose, Amites Dasgupta, and Krishanu Maulik

#### Abstract

Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak limits are known.

#### Article information

**Source**

Bernoulli, Volume 15, Number 1 (2009), 279-295.

**Dates**

First available in Project Euclid: 3 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1233669892

**Digital Object Identifier**

doi:10.3150/08-BEJ150

**Mathematical Reviews number (MathSciNet)**

MR2546808

**Zentralblatt MATH identifier**

1206.60008

**Keywords**

martingale reducible stochastic replacement matrix urn model variance mixture of normal

#### Citation

Bose, Arup; Dasgupta, Amites; Maulik, Krishanu. Multicolor urn models with reducible replacement matrices. Bernoulli 15 (2009), no. 1, 279--295. doi:10.3150/08-BEJ150. https://projecteuclid.org/euclid.bj/1233669892