Concentration and Moment Inequalities for Polynomials of Independent Random Variables

TL;DR

This paper introduces a versatile method for establishing moment and concentration inequalities for polynomials of diverse independent random variables, addressing previously intractable cases and improving upon existing bounds.

Contribution

The authors develop a general approach applicable to many types of random variables, enabling new concentration bounds for high-degree polynomials and solving open problems like the permanent of random matrices.

Findings

Method applies to Gaussian, Boolean, exponential, Poisson variables

Yields stronger concentration inequalities than Kim and Vu

Proves tight bounds for Boolean variables

Abstract

In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of a random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due to Kim and Vu. The main advantage of our method in comparison with the existing ones is a wide range of random variables we can handle and bounds for previously intractable regimes of high degree polynomials and small expectations. On the negative side we show that even for boolean random variables each term in our concentration inequality is tight.