Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns
نویسندگان
چکیده
The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set ofminimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.
منابع مشابه
A Central Limit Theorem and Its Applications to Multicolor Randomly Reinforced Urns
Let (Xn) be a sequence of integrable real random variables, adapted to a filtration (Gn). Define
متن کاملNecklace Processes via Pólya Urns
Mallows and Shepp (2008) developed the following necklace processes. Start with a necklace consisting of one white bead and one black bead, and insert, one at a time, under a deterministic rule, a white bead or a black bead between a randomly chosen adjacent pair. They studied the statistical properties of the number of white beads by investigating the nature of the moments and the expected num...
متن کاملFunctional Limit Theorems for Multitype Branching Processes and Generalized Pólya Urns
A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the Athreya–Karlin embedding, these results yield asymptotic results for generalized Pólya urns. We investigate such results in...
متن کاملCentral limit theorems for a hypergeometric randomly reinforced urn
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color given the past is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence ...
متن کاملClassification of large Pólya-Eggenberger urns with regard to their asymptotics
This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having “large” eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn’s replacement matrix and examples of each case are treated. We study the ca...
متن کامل