Stein Couplings, Log Concavity and Concentration of Measure

Stochastics Seminar
Tuesday, May 19, 2015 - 15:05
1 hour (actually 50 minutes)
Skiles 005
University of Minnesota
For a nonnegative random variable Y with finite nonzero mean \mu, we say that Y^s has the Y-size bias distribution if E[Yf(Y)] = \mu E[f(Y^s)] for all bounded, measurable f. If Y can be coupled to Y^s having the Y-size bias distribution such that for some constant C we have Y^s \leq Y + C, then Y satisfies a 'Poisson tail' concentration of measure inequality. This yields concentration results for examples including urn occupancy statistics for multinomial allocation models and Germ-Grain models in stochastic geometry, which are members of a class of models with log concave marginals for which size bias couplings may be constructed more generally. Similarly, concentration bounds can be shown when one can construct a bounded zero bias coupling or a Stein pair for a mean zero random variable Y. These latter couplings can be used to demonstrate concentration in Hoeffding's permutation and doubly indexed permutations statistics. The bounds produced, which have their origin in Stein's method, offer improvements over those obtained by using other methods available in the literature. This work is joint with J. Bartroff, S. Ghosh and L. Goldstein.