Azuma-Hoeffding bounds for a class of urn models

TL;DR

This paper derives Azuma-Hoeffding bounds for urn models by relating variables to eigenvectors and martingales, including cases with repeated eigenvalues, to analyze deviations from the limit.

Contribution

It introduces a novel method to obtain concentration bounds for urn models using eigenvector-based martingales, extending to cases with repeated eigenvalues.

Findings

Established Azuma bounds for a class of urn models.

Extended analysis to cases with repeated eigenvalues.

Provided a framework for probabilistic deviation bounds in urn processes.

Abstract

We obtain Azuma bounds for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using Jordan vectors.