Oscillations for order statistics of some discrete processes

TL;DR

This paper investigates the oscillatory behavior of the maximum number of balls in boxes in a large uniform allocation, revealing a periodicity and distributional properties linked to the tail of the underlying discrete distribution.

Contribution

It demonstrates that the maximum in such discrete processes oscillates between two values with a periodic pattern, and characterizes the limiting distribution based on the tail behavior.

Findings

Maximum takes at most two values with high probability

Oscillations depend on the tail of the distribution

Results apply to various allocation problems

Abstract

Suppose $k$ balls are dropped into $n$ boxes independently with uniform probability, where $n, k$ are large with ratio approximately equal to some positive real $\lambda$. The maximum box count has a counterintuitive behavior: first of all, with high probability it takes at most two values $m_n$ or $m_n+1$, where $m_n$ is roughly $\frac{\ln n}{\ln \ln n}$. Moreover, it oscillates between these two values with an unusual periodicity. In order to prove this statement and various generalizations, it is first shown that for $X_1,...,X_n$ independent and identically distributed discrete random variables with common distribution $F$, under mild conditions, the limiting distribution of their maximum oscillates in three possible families, depending on the tail of the distribution. The result stated at the beginning follows from the equivalence of ensemble for the order statistics in various allocations problems, obtained via conditioning limit theory. Results about the number of ties for the maximum, as well as applications, are also provided.