Time-dependent Pólya urn

Abstract

We consider a time-dependent version of a Pólya urn containing black and white balls. At each time $n$ a ball is drawn from the urn at random and replaced in the urn along with ${\sigma }_n$ additional balls of the same colour. The proportion of white balls converges almost surely to a random limit $\Theta$, and $\mathcal{D}=\{\Theta \in \{0,1\}\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $({\sigma }_n)$, between the regimes $\mathbb{P}(\mathcal{D})=1$ and $\mathbb{P}(\mathcal{D})<1$ was obtained by R. Pemantle in 1990. We describe the phase transition between the cases $\mathbb{P}(\mathcal{D})=0$ and $\mathbb{P}(\mathcal{D})>0$. Further, we study the stronger monopoly event $\mathcal{M}$ when one of the colours eventually stops reappearing, and analyse the phase transition between the regimes $\mathbb{P}(\mathcal{M})=0$, $\mathbb{P}(\mathcal{M})\in (0,1)$, and $\mathbb{P}(\mathcal{M})=1$.

References in corpus (1)

• A survey of random processes with reinforcement

Cited by in corpus (1)

• Time-dependent balls and bins model with positive feedback