Concentration inequalities via zero bias couplings

Abstract

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.

Size biased couplings and the spectral gap for random regular graphs

“…Previous work for concentration by unbounded size biased couplings was limited to [GGR11], with a construction particular to the example treated, and dependent on a negative association property holding. There was no previous Bennett-type inequality by size biasing, though [GI14] gives one by the related method of zero biasing; see Remark 5.2.…”

Exponential bounds for the hypergeometric distribution

“…for all λ > 0. Goldstein and Işlak [6] recently used a variant of Stein's method to give another inequality for the tails of Hoeffding's statistic S:…”

The Bennett-Orlicz Norm

“…But the recent developments of concentration inequalities via Stein’s method yields inequalities of the form given in Eq. 2.7 for many random variables Z which are not sums of independent random variables: see, for example, Ghosh and Goldstein (2011a), Ghosh and Goldstein (2011b), & Goldstein and Iṡlak (2014). The point of the previous proposition is that (up to constants) these inequalities in terms of h 0 can be re-expressed in terms of the (larger) function h 1 .…”