# Stein Couplings, Log Concavity and Concentration of Measure

Thursday, April 9, 2015 - 2:00pm - 3:00pm

Lind 305

Larry Goldstein (University of Southern California)

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

These methods yield 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 of a mean zero random variable Y with finite nonzero variance \sigma^2 to a Y^* satisfying

E[Yf(Y)] = \sigma^2 E[f'(Y^*)] for all smooth f.

These couplings can be used to demonstrate concentration in Hoeffding's combinatorial central limit theorem under diverse assumptions on the permutation distribution.

The bounds produced by these couplings, 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 Jay Bartroff and Umit Islak.

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

These methods yield 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 of a mean zero random variable Y with finite nonzero variance \sigma^2 to a Y^* satisfying

E[Yf(Y)] = \sigma^2 E[f'(Y^*)] for all smooth f.

These couplings can be used to demonstrate concentration in Hoeffding's combinatorial central limit theorem under diverse assumptions on the permutation distribution.

The bounds produced by these couplings, 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 Jay Bartroff and Umit Islak.