On Sharpest Tail Bounds for Functions of Tail Bounded Random Variables

TL;DR

This paper investigates the exact determination of sharpest tail bounds for functions of tail-bounded random variables, addressing a largely unexplored question in probability theory.

Contribution

It introduces methods to find the exact sharpest tail bounds in special cases and reduces the search space for dependent and independent variables.

Findings

Exact tail bounds are found in some special cases.

Reductions in the space of random variables are developed for both independent and dependent cases.

Closed-form solutions for the sharpest bounds are rarely available.

Abstract

Consider $n$ real/complex, independent/dependent random variables with respective tail bounds and $g$ a measurable function of the r.v.'s. Consider $f$ the "sharpest" tail bound of $g$ (sharpest in the sense that if $f$ were any less, then for some $X_1,...,X_n$ satisfying the conditions, $g(X_1,...,X_n)$ would not satisfy $f$). Significant research has been done to approximate $f$ often with high accuracy. These results are often of the form that for $g$ in this family and tail bounds of $X_k$ in this family, $f$ is bounded by some $f'$ with high accuracy. However, the question "what would it take to find $f$ exactly?" has received little attention, apparently even for simple cases. This is the question we try to answer. For $X_1,...,X_n$ required to be mutually independent, first the $X_k$ are simplified to be monotone on $(0,1)$ WLOG. This strengthens convergence in distribution to convergence a.e. (Skorokhod's representation theorem) and allows defining shift operators, which help reduce the space of r.v.'s one searches to find $f$ and/or the maximum measure of a subset. We do find $f$ in some special cases, however $f$ rarely has a closed form. For $X_1,...,X_n$ dependent/not necessarily independent, another reduction in the space of r.v.'s one searches to find $f$ is done.