Statistics Papers

Document Type

Journal Article

Date of this Version

2014

Publication Source

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Volume

50

Issue

4

Start Page

1121

Last Page

1139

DOI

10.1214/13-AIHP557

Abstract

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand’s lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini–Kalai–Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand’s bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.

Keywords

influences, geometric influences, noise sensitivity, correlation between increasing sets, Talagrand’s bound, Gaussian measure, isoperimetric inequality

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Date Posted: 27 November 2017

This document has been peer reviewed.