Concentration inequalities via zero bias couplings

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.