Large deviation inequalities for the nonlinear unbalanced urn model

TL;DR

This paper establishes large deviation inequalities for a two-color nonlinear unbalanced urn model using stochastic approximation theory, providing new bounds and insights into the model's probabilistic behavior.

Contribution

It introduces novel large deviation inequalities for nonlinear unbalanced urn models and extends stochastic approximation theory to this context.

Findings

Derived large deviation bounds for the urn model

Provided a general large deviation inequality for stochastic approximation algorithms

Enhanced understanding of probabilistic behavior in nonlinear urn processes

Abstract

In the present paper, we consider the two-color nonlinear unbalanced urn model, under a drawing rule reinforced by an $\mathbb{R}^+$-valued concave function and an unbalanced replacement matrix. The large deviation inequalities for the nonlinear unbalanced urn model are established by using the stochastic approximation theory. As an auxiliary theory, we give a specific large deviation inequality for a general stochastic approximation algorithm.