Asymptotics for randomly reinforced urns with random barriers

E-ISSN： 1475-6072|53|4|1206-1220

ISSN： 0021-9002

Source： Journal of Applied Probability, Vol.53, Iss.4, 2016-12, pp. : 1206-1220

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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.