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≤L<U≤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 that R_n=B_n. Then, under mild conditions, it is shown that Z_n→^a.s.Z for some random variable Z, and D_n≔ √n(Z_n-Z) →𝒩(0,σ²) conditionally almost surely (a.s.), where σ² is a certain random variance. Almost sure conditional convergence means that ℙ(D_n∈⋅|𝒢_n) →^w 𝒩(0,σ²) a.s., where ℙ(D_n∈⋅|𝒢_n) is a regular version of the conditional distribution of D_n given the past 𝒢_n. Thus, in particular, one obtains D_n→𝒩(0,σ²) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.