Bounds on the Poincare constant under negative dependence

TL;DR

This paper establishes bounds on the Poincare constant for negatively dependent, non-negative integer-valued variables, with applications to various probabilistic models and Poisson convergence.

Contribution

It provides new bounds on the Poincare constant under negative dependence, improving understanding of spectral gaps in dependent discrete distributions.

Findings

Bounds compare favorably with existing results

Applications demonstrated in occupancy and urn models

Supports Poisson convergence theorems

Abstract

We give bounds on the Poincare (inverse spectral gap) constant of a non-negative, integer-valued random variable W, under negative dependence assumptions such as ultra log-concavity and total negative dependence. We show that the bounds obtained compare well to others in the literature. Examples treated include some occupancy and urn models, a random graph model and small spacings on the circumference of a circle. Applications to Poisson convergence theorems are considered.