Applications of size biased couplings for concentration of measures

Let Y be a nonnegative random variable with mean μ and finite positive variance σ, and let Y , defined on the same space as Y , have the Y size biased distribution, that is, the distribution characterized by E[Y f(Y )] = μEf(Y ) for all functions f for which these expectations exist. Under a variety of conditions on the coupling of Y and Y , including combinations of boundedness and monotonicity, concentration of measure inequalities such as P ( Y − μ σ ≥ t ) ≤ exp ( − t 2 2(A+Bt) ) for all t ≥ 0 were shown to hold for some explicit A and B in [14], and were applied to the lightbulb process of [34]. Those concentration of measure results are applied here to a number of new examples where one can construct a bounded coupling of Y s to Y : the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m-runs in a sequence of coin tosses, the number of local maxima of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, and the volume covered by the union of n balls placed uniformly over a volume n subset of R. Concentration of measure results are also provided for certain instances where the coupling of Y s to Y is not bounded, in particular, for the number of isolated vertices in the ErdösRényi random graph model, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition, the latter with applications to the generalized variance of a random matrix.