An urn contains black and red balls. Let Zn be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball bn is drawn. If bn is black and Zn-1<U, then bn is replaced together with a random number Bn of black balls. If bn is red and Zn-1>L, then bn is replaced together with a random number Rn of red balls. Otherwise, no additional balls are added, and bn alone is replaced. In this paper we assume that Rn=Bn. Then, under mild conditions, it is shown that Zn→a.s.Z for some random variable Z, and Dn≔ √n(Zn-Z) →𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(Dn∈⋅|𝒢n) →w 𝒩(0,σ2) a.s., where ℙ(Dn∈⋅|𝒢n) is a regular version of the conditional distribution of Dn given the past 𝒢n. Thus, in particular, one obtains Dn→𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.
"Asymptotics for randomly reinforced urns with random barriers." J. Appl. Probab. 53 (4) 1206 - 1220, December 2016.