On maximal tail probability of sums of nonnegative, independent and identically distributed random variables

TL;DR

This paper derives optimal upper bounds for the tail probability of sums of nonnegative i.i.d. random variables, extending known results for small sums to general cases using extremal graph theory techniques.

Contribution

It provides the first solution for the tail probability bounds for sums of three or more i.i.d. nonnegative variables under certain conditions, linking probability bounds to extremal hypergraph theory.

Findings

Solved for k=3 and x ≤ 1/(2k-1)

Extended tail probability bounds to general k

Connected probability bounds with extremal hypergraph theory

Abstract

We consider the problem of finding the optimal upper bound for the tail probability of a sum of $k$ nonnegative, independent and identically distributed random variables with given mean $x$. For $k=1$ the answer is given by Markov's inequality and for $k=2$ the solution was found by Hoeffding and Shrikhande in 1955. We solve the problem for $k=3$ as well as for general $k$ and $x\leq1/(2k-1)$ by showing that it follows from the fractional version of an extremal graph theory problem of Erd\H{o}s on matchings in hypergraphs.