  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.
