Gaussian process approximations for multicolor P\'olya urn models

TL;DR

This paper develops Gaussian process approximations for multicolor Pólya urn models, providing insights into their long-term behavior and deviations, especially relevant for tissue growth modeling.

Contribution

It introduces a novel Gaussian approximation framework for multicolor Pólya urns using strong approximation theorems, accounting for large initial sizes and long-term dynamics.

Findings

Gaussian process approximations for urn dynamics

Deviation bounds for urn composition distributions

Dependence of approximation on initial size and steps

Abstract

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor P\'olya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the P\'olya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. Which of the two terms dominates depends on the ratio of the number of time steps $n$ to the initial number of balls $N$ in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step $n$ urn composition distribution from the approximating normal law.