Concentration inequalities via zero bias couplings

TL;DR

This paper derives new concentration inequalities for mean-zero random variables using zero bias couplings, applicable even when variables are not independent, with potential applications to permutation-based statistics.

Contribution

It introduces novel concentration bounds based on zero bias couplings, extending measure concentration results to dependent variables like permutation statistics.

Findings

Provides explicit tail bounds for zero biased variables.

Extends concentration inequalities to dependent structures such as permutation statistics.

Applicable under conditions on the moment generating function.

Abstract

The tails of the distribution of a mean zero, variance $\sigma^2$ random variable $Y$ satisfy concentration of measure inequalities of the form $\mathbb{P}(Y \ge t) \le \exp(-B(t))$ for $$ B(t)=\frac{t^2}{2( \sigma^2 + ct)} \quad \mbox{for $t \ge 0$, and} \quad B(t)=\frac{t}{c}\left( \log t - \log \log t - \frac{\sigma^2}{c}\right) \quad \mbox{for $t>e$} $$ whenever there exists a zero biased coupling of $Y$ bounded by $c$, under suitable conditions on the existence of the moment generating function of $Y$. These inequalities apply in cases where $Y$ is not a function of independent variables, such as for the Hoeffding statistic $Y=\sum_{i=1}^n a_{i\pi(i)}$ where $A=(a_{ij})_{1 \le i,j \le n} \in \mathbb{R}^{n \times n}$ and the permutation $\pi$ has the uniform distribution over the symmetric group, and when its distribution is constant on cycle type.