TALK by L. Goldstein

Souther California Probability Symposium

December 3, 2005

Zero Biasing and Combinatorial Central Limit Theorems

Larry Goldstein (USC)

For a mean zero, variance one random variable W, we say W^* has the W zero biased distribution if

EWf(W) = Ef'(W^*) for all smooth f.

It can be shown using Stein's method, that W, with distribution F, is close to normal when it can be coupled closely to its zero biasd version W^* with distribution F^*, as quantified by the L¹ norm inequality

||F - Phi||₁ <= 2||F^* - F||₁,

where Phi is the cumulative standard normal. The bound provides Berry-Esseen type inequalities, with explicit, moderate, constants. By the use of smoothing inequalities, L^infinity bounds can also be derived in terms of distances between W and W^*. Bounds of these types will be illustrated to assess the quality of the normal approximation in combinatorial central limit theorems; that is, to

Y = sum_i=1,2,...,n a_{i, pi(i)}

when the random permutation pi has distribution uniform over the symmetric group S_n, and also for certain distributions on S_n which are constant on conjugacy classes.

This abstract is available as a pdf file and as a dvi file.