Concentration of measures via size biased couplings

Abstract

Let $Y$ be a nonnegative random variable with mean $μ$ and finite positive variance $σ^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by E[Yf(Y)]=μE f(Y^s) for all functions $f$ for which these expectations exist. Under a variety of conditions on the coupling of Y and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities hold. Examples include 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 maximum of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, the volume covered by the union of $n$ balls placed uniformly over a volume n subset of d dimensional Euclidean space, the number of bulbs switched on at the terminal time in the so called lightbulb process, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition.

Concentration results for the number of isolated vertices have been removed from this version, and with corrections, posted jointly with Martin Raic in http://arxiv.org/abs/1106.0048