Statistics Papers

Document Type

Journal Article

Date of this Version

2012

Publication Source

The Annals of Probability

Volume

40

Issue

3

Start Page

1135

Last Page

1166

DOI

10.1214/11-AOP643

Abstract

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn–Kalai–Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in ℝn of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least ct(1 − t)√log n/n, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝn and the class of sets invariant under transitive permutation group of the coordinates.

Keywords

influences, product space, Kahn-Kalai-Linial influence bound, Gaussian measure, isoperimetric inequality

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Date Posted: 27 November 2017

This document has been peer reviewed.