Statistics Papers

Document Type

Journal Article

Date of this Version

4-2013

Publication Source

Arkiv för Matematik

Volume

51

Issue

1

Start Page

29

Last Page

52

DOI

10.1007/s11512-011-0145-5

Abstract

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by δ are called δ-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

In our main result we show that E⁡[f1(X11,…,X1n)…fk(Xk1,…,Xkn)] is close to 0 under the following assumptions:

  • the vectors{(X1j,…,Xkj) : 1 ≤ j ≤ n} are independent identically distributed, and for each j the vector (X1j,…,Xkj) has a pairwise independent distribution.

  • the functions fi are uniform;

  • the functions fi are of low degree.

We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s11512-011-0145-5.

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Date Posted: 27 November 2017

This document has been peer reviewed.