Asymptotics for randomly reinforced urns with random barriers. (9th December 2016)

• Record Type: Journal Article
• Authors: Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro
• Is Part Of: Journal of applied probability. Volume 53:Number 4(2016)
• Page Start: 1206
• Page End: 1220
• Publication Date: 2016-12-09
• Subjects: Bayesian nonparametric, -- central limit theorem, -- random probability measure, -- stable convergence, -- urn model; Primary 60B10, -- Secondary 60F05, -- 60G57, -- 62F15; 519.2
• DOI: 10.1017/jpr.2016.75
• Languages: English
• ISSNs: 0021-9002

Abstract

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, σ 2 ) conditionally almost surely (a.s.), where σ 2 is a certain random variance. Almost sure conditional convergence means that ℙ( D n ∈⋅|𝒢 n )→ w 𝒩(0, σ 2 ) 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, σ 2 ) stably. It is also shown that L < Z < U a.s. and Z has nonatomic distribution.