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Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry and multinomial allocation models. The results obtained compare favorably with classical methods, including the use of McDiarmid's inequality, negative association, and self bounding functions.

The tails of the distribution of a mean zero, variance σ 2 random variable Y satisfy concentration of measure inequalities of the form P(Y ≥ t) ≤ exp(−B(t)) forfor t ≥ 0, and B(t) = t c log t − log log t − σ 2 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 = n i=1 a iπ(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.

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