Asymptotics for randomly reinforced urns with random barriers

Abstract

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\le L<U\le 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\stackrel{a.s.}{⟶}Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,σ^2)\quad\text{conditionally a.s.} \end{gather*} where ${\sigma }^2$ is a certain random variance. Almost sure conditional convergence means that \begin{gather*} P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,σ^2)\quad\text{a.s.} \end{gather*} where $P(D_n\in \cdot \mid {\mathcal{G}}_n)$ is a regular version of the conditional distribution of $D_n$ given the past ${\mathcal{G}}_n$. Thus, in particular, one obtains $D_n⟶\mathcal{N}(0,{\sigma }^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.

13 pages, submitted