Chebyshev Inequalities for Products of Random Variables

TL;DR

This paper develops sharp Chebyshev bounds for the tail probabilities of products of symmetric non-negative random variables using only their first two moments, with methods applicable to both precise and imprecise covariance information.

Contribution

It introduces novel bounds for tail probabilities of products, applicable under exact or bounded covariance, and analyzes their behavior for large numbers of variables.

Findings

Bounds are computable via semidefinite programming with exact covariance.

Analytic bounds are available when only an upper bound on covariance is known.

Left tail bounds become trivial (equal to 1) when the number of variables exceeds a threshold.

Abstract

We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is available, the probability bounds on the right tails can be evaluated analytically. The bounds under precise and imprecise covariance information coincide for all left tails as well as for all right tails corresponding to quantiles that are either sufficiently small or sufficiently large. We also prove that all left probability bounds reduce to the trivial bound 1 if the number of random variables in the product exceeds an explicit threshold. Thus, in the worst case, the weak-sense geometric random walk defined through the running product of the random variables is absorbed at 0 with certainty as soon as time exceeds the given threshold. The techniques devised for constructing Chebyshev bounds for products can also be used to derive Chebyshev bounds for sums, maxima and minima of non-negative random variables.