Asymptotics for randomly reinforced urns with random barriers

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 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, σ) conditionally a.s. where σ is a certain random variance. Almost sure conditional convergence means that P ( Dn ∈ · | Gn ) weakly −→ N (0, σ) 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, σ) stably. It is also shown that L < Z < U a.s. and Z has non-atomic distribution.