On generalized P\'olya urn models

TL;DR

This paper extends the classical Pólya urn model to multiple balls drawn per step and multiple colors, providing exact formulas for moments and exploring new sampling schemes and generalized models.

Contribution

It introduces a generalized urn model with multiple balls drawn at each step, derives exact expectation and variance formulas, and discusses extensions to multiple colors and new sampling methods.

Findings

Exact expressions for expectation and variance of white balls after n draws.

Structural analysis of higher moments in the generalized urn model.

Introduction of new urn models with step-by-step sampling and multiple colors.

Abstract

We study an urn model introduced in the paper of Chen and Wei, where at each discrete time step $m$ balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known P\'olya-Eggenberger urn model, case m=1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.