Decoupling inequalities with exponential constants

TL;DR

This paper establishes geometric conditions under which decoupling inequalities for vector-valued polynomials have constants that grow exponentially with the polynomial's degree, enhancing understanding of dependence structures in random objects.

Contribution

It provides new geometric criteria for decoupling inequalities with exponential constants, applicable to Banach spaces with finite cotype and stronger geometric assumptions.

Findings

Decoupling inequalities with exponential constants are achievable under certain geometric conditions.

The results extend to inequalities involving only two independent copies of the random vector.

New bounds are provided for classical decoupling inequalities in Banach spaces.

Abstract

Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring decoupling inequalities with good constants depending only exponentially on the degree of the polynomial. Assuming the Banach space has finite cotype we achieve this for classical decoupling inequalities that compare the polynomials with their associated multilinear operators. Under stronger geometric assumptions on the involved Banach spaces, we also obtain decoupling inequalities between random polynomials and fully independent random sums of their coefficients. Finally, we present decoupling inequalities where in the multilinear operator just two independent copies of the random vector are involved (one repeated $m-1$ times).