A Gaussian Process Approximation for a two-color Randomly Reinforced Urns

TL;DR

This paper develops a Gaussian process approximation for the composition of a two-color randomly reinforced urn, enabling the derivation of limit theorems and distributional properties under finite moment conditions.

Contribution

It introduces a Gaussian approximation framework for reinforced urns with equal and unequal reinforcement means, extending the theoretical understanding of their asymptotic behavior.

Findings

Established Gaussian process approximation for urn compositions

Proved law of the iterated logarithm and functional limit theorems

Showed the urn composition distribution has no point masses under finite moments

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.