Abstract
We prove a Gaussian process approximation for the sequence of random compositions of a two-color randomly reinforced urn for both the cases with the equal and unequal reinforcement means. By using the Gaussian approximation, the law of the iterated logarithm and the functional limit central limit theorem in both the stable convergence sense and the almost-sure conditional convergence sense are established. Also as a consequence, we are able to to prove that the distribution of the urn composition has no points masses both when the reinforcement means are equal and unequal under the assumption of only finite $(2+\epsilon)$-th moments.
Highlights
Asymptotic properties, including the strong consistency and asymptotic normality, of urn models and their applications are widely studied in recent years under various assumptions concerning the updating rules, for example, one may refer to Chauvin, Pouyanne and Sahnoun (2011), Bai, Hu and Rosenberger (2002), Bai and Hu (2005), Hu and Rosenberger (2006), Janson (2004, 2006), Zhang, Hu and Cheung (2006) etc
We consider a kind of two-color urn model, called the randomly reinforced urn (RRU) model, which is a generalization of the original Pólya urn
For deriving the asymptotic distributions, the approximations (2.6) and (3.4) seem more powerful than (2.1) and (3.1) because the process for approximation is independent of other random variables considered. (2.1) and (3.1) are helpful for establishing the strong convergence such as the law of the iterated logarithm
Summary
Asymptotic properties, including the strong consistency and asymptotic normality, of urn models and their applications are widely studied in recent years under various assumptions concerning the updating rules, for example, one may refer to Chauvin, Pouyanne and Sahnoun (2011), Bai, Hu and Rosenberger (2002), Bai and Hu (2005), Hu and Rosenberger (2006), Janson (2004, 2006), Zhang, Hu and Cheung (2006) etc. The sampled ball is replaced in the urn together with a nonnegative random number Um+1,k of balls of the same type k, generated from a distribution μk with mean mk > 0. This is the model introduced and formally named the randomly reinforced urn in Muliere, Paganoni and Secchi (2006a). When m1 = m2) under the assumption that μ1 and μ2 have only finite (2 + )-th moments Another implication of our Gaussian approximation is that we are able to establish the central limit theorem in a simple way for the random number Nn,k of draws, where Nn,k is the number of type k balls being drawn in the first n samplings.
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