Balls and Bins -- Simple Concentration Bounds

TL;DR

This paper derives simple concentration bounds for the distribution of loads in balls-and-bins models, providing insights into maximum load and waiting times based on probabilistic inequalities.

Contribution

It introduces new concentration bounds for balls-and-bins processes that relate load probabilities to distribution norms, enhancing understanding of load distribution.

Findings

Bounds for maximum load probabilities

Bounds for waiting times until k-loaded bins

Control of load distribution via distribution norms

Abstract

Concentration bounds are given for throwing balls into bins independently according to a distribution $p$. The probability of a $k$-loaded bin after $m$ balls is shown to be controlled on both sides by $\rho_{m,k} := m \|p\|_k / k$. This gives concentration inequalities for the maximum load as well as for the waiting time until a $k$-loaded bin.