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

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 ≤ 1/(2k − 1) by showing that it follows from the fractional version of an extremal graph theory problem of Erdős on matchings in hypergraphs.

Uniformity-independent minimum degree conditions for perfect matchings in hypergraphs

“…Although our proof of this theorem does not involve any significant new ideas -indeed, after this note first appeared on the arXiv, we were informed by Andrey Kupavskii that such an improvement using the same methods is also implicit in his work [5] with Peter Frankl on the Erdős Matching Conjecture -we believe that the statement itself is important enough to be worth isolating and highlighting. It was also pointed out to us by Andrey Kupavskii that the equivalence between the problems established in Theorem 1.14 already appears (with the same proof) in the work of Łuczak, Mieczkowska, and Šileikis [11], although in this work, no mention is made either of Feige's conjecture, or the implication for Dirac-type thresholds for perfect matchings in hypergraphs.…”

“…Acknowledgement. We would like to thank Mathias Schacht for communicating this problem to us, and Andrey Kupavskii for bringing references [5,11] to our attention.…”

A Generalization of Erdős' Matching Conjecture

“…The current best known upper bound on this constant is c r ≤ 2r + 1, and is due to Frankl [7]. Let us also remark that Erdős' matching conjecture, if true, has implications in game theory (see [5]), distributed storage allocation (see [3,Section 5]) as well as in probability theory (see [3,14]).…”

The Erdős Matching Conjecture and concentration inequalities

“…We note that it is easy to see that m k (x) = 1 for x ≥ 1/k. The authors of [39] proved the equivalence of ( 14) and ( 8), which implied that ( 14) is true for k = 3 and any x, as well as for any k and x ≤ 1 2k−1 . Theorem 1, combined with the aforementioned equivalence, immediately implies the following corollary.…”

“…The value of p 2 (x) was determined by Hoeffding and Shrikhande [28]. Luczak, Mieczkowska and Šileikis [39] proposed the following conjecture, which states that for every positive k and 0…”