Asymptotics for randomly reinforced urns with random barriers

Publisher: Cambridge University Press

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.