Talagrand inequality at second order and application to Boolean analysis

2018/01/26 by Tanguy, Kevin
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.08931

Abstract

This note is concerned with an extension, at second order, of an inequality on the discrete cube C_n=\-1,1\ (equipped with the uniform measure) due to Talagrand (\citeTalL1L2). As an application, the main result of this note is a Theorem in the spirit of a famous result from Kahn, Kalai and Linial (cf. \citeKKL) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates (i,j)∈\1,…,n\² is introduced in section 2 and the following alternative is obtained : for any Boolean function f : C_n→ \0,1\, either there exists a coordinate with influence at least of order (1/n)^1/(1+η), with 0

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