Concentration of measures via size biased couplings

TL;DR

This paper develops concentration of measure inequalities for nonnegative random variables using size biased couplings, with applications to combinatorics, geometry, and probability distributions.

Contribution

It introduces new concentration inequalities based on size biased couplings under various boundedness and monotonicity conditions.

Findings

Establishes concentration bounds for various combinatorial and geometric models.

Applies to distributions like infinitely divisible and compound Poisson with bounded moment generating functions.

Demonstrates the versatility of size biased couplings in deriving measure concentration.

Abstract

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, 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)]=\mu E 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.