Concentration inequalities via zero bias couplings

The tails of the distribution of a mean zero, variance σ² random variable Y satisfy concentration of measure inequalities of the form P(Y≥t) ≤ exp(-B(t)) for B(t)=t²/2(σ² + ct) for t ≥ 0, and B(t)=t/c(log t -log log t-σ²/c)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=∑_i=1ⁿaiπ(i) where A=(a_ij)_1≤i,j≤n ∈R^n×n and the permutation π has the uniform distribution over the symmetric group, and when its distribution is constant on cycle type.