Asymptotics for randomly reinforced urns with random barriers. (English) Zbl 1358.60005

Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro

J. Appl. Probab. 53, No. 4, 1206-1220 (2016).

Summary: An urn contains black and red balls. Let \(Z_{n}\) be the proportion of black balls at time \(n\) and \(0\leq L <U\leq 1\) 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 \(Z_{n}\overset{\text{a.s.}}{\longrightarrow} Z\) for some random variable \(Z\), and \(D_{n} := \sqrt{n}(Z_{n}-Z)\to \mathcal{N}(0,\sigma^{2})\) conditionally almost surely (a.s.), where \(\sigma^{2}\) is a certain random variance. Almost sure conditional convergence means that \(\mathbb{P}(D_{n} \in \cdot | \mathcal{G}_{n}) \overset{\text{a.s.}}{\longrightarrow} \mathcal{N}(0,\sigma^{2})\) a.s., where \(\mathbb{P}(D_{n} \in \cdot| \mathcal{G}_{n})\) is a regular version of the conditional distribution of \(D_{n}\) given the past \(\mathcal{G}_{n}\). Thus, in particular, one obtains \(D_{n} \to \mathcal{N}(0,\sigma^{2})\) stably. It is also shown that \(L<Z<U\) a.s. and \(Z\) has nonatomic distribution.

MSC:

60B10	Convergence of probability measures
60F05	Central limit and other weak theorems
60G57	Random measures
62F15	Bayesian inference

Keywords:

Bayesian nonparametric; central limit theorem; random probability measure; stable convergence; urn model