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 $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,\sigma^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,\,\sigma^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,\sigma^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.

Asymptotics for randomly reinforced urns with random barriers

RIGO, PIETRO
2016

Abstract

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\le LL$, 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 $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,\sigma^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,\,\sigma^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,\sigma^2)$ stably. It is also shown that $L
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11571/1113710
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact