Asymptotics for randomly reinforced urns with random barriers

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0le LL$, 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 $Z_noverseta.s.longrightarrow Z$ for some random variable $Z$, and egingather* D_n:=sqrtn,(Z_n-Z)longrightarrowmathcalN(0,sigma^2) quad extconditionally a.s. endgather* where $sigma^2$ is a certain random variance. Almost sure conditional convergence means that egingather* Pigl(D_nincdotmidmathcalG_nigr)oversetweaklylongrightarrow mathcalN(0,,sigma^2)quad exta.s. endgather* where $Pigl(D_nincdotmidmathcalG_nigr)$ is a regular version of the conditional distribution of $D_n$ given the past $mathcalG_n$. Thus, in particular, one obtains $D_nlongrightarrowmathcalN(0,sigma^2)$ stably. It is also shown that $L