Talagrand Inequality at Second Order and Application to Boolean Analysis

Kevin Tanguy ()
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Kevin Tanguy: University of Angers

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 692-714

Abstract: Abstract This note is concerned with an extension, at second order, of an inequality on the discrete cube $$C_n=\{-\,1,1\}^n$$Cn={-1,1}n (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates $$(i,j)\in \{1,\ldots ,n\}^2$$(i,j)∈{1,…,n}2 is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function $$f\,:\, C_n\rightarrow \{0,1\}$$f:Cn→{0,1}, either there exists a coordinate with influence at least of order $$(1/n)^{1/(1+\eta )}$$(1/n)1/(1+η), with $$\, 0

Keywords: Boolean analysis; Influences; Hypercontractivity; Functional inequalities; 60E15; 60J25; 06E30; 28A12 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00957-2

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