The lower tail: Poisson approximation revisited

TL;DR

This paper revisits Poisson approximation for the lower tail probabilities of sums of dependent indicators, establishing optimal bounds for large deviations and introducing correlation-based methods for broader cases.

Contribution

It demonstrates the optimality of Janson's inequality for large deviations when the sum is approximately Poisson and introduces correlation-based approaches for non-Poisson cases.

Findings

Janson's inequality is optimal for large deviations in approximately Poisson sums.

New lower tail estimates for subgraph counts in random graphs.

Correlation-based methods extend results beyond the Poisson approximation.

Abstract

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case \eps=1) of Janson, Luczak and Rucinski.