Urn models with random multiple drawing and random addition

TL;DR

This paper introduces a highly general urn model with random multiple sampling and time-dependent random additions, providing convergence results, fluctuation theorems, and confidence intervals for the proportion of colors.

Contribution

It extends previous urn models by allowing random sample sizes and general random addition matrices, with rigorous convergence and fluctuation analysis.

Findings

Almost sure convergence of color proportions

Central limit theorems for fluctuations

Asymptotic confidence intervals for proportions

Abstract

We consider an urn model with multiple drawing and random time-dependent addition matrix. The model is very general with respect to previous literature: the number of sampled balls at each time-step is random, the addition matrix has general random entries. For the proportion of balls of a given color, we prove almost sure convergence results and fluctuation theorems (through CLTs in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution). Asymptotic confidence intervals are given for the limit proportion, whose distribution is generally unknown.