A phase transition for the probability of being a maximum among random vectors with general iid coordinates

TL;DR

This paper investigates the probability that a specific vector is a maximum among iid vectors with general distributions, revealing a phase transition depending on the growth rate of vector dimension relative to the sample size.

Contribution

It characterizes the phase transition of the maximum probability in high dimensions for vectors with general iid coordinates, extending previous results to broader distributions.

Findings

Probability tends to zero if dimension grows faster than a logarithmic factor.

Probability tends to one if dimension grows slower than that factor.

The phase transition rate is a functional of the distribution function F.

Abstract

Consider $n$ iid real-valued random vectors of size $k$ having iid coordinates with a general distribution function $F$. A vector is a maximum if and only if there is no other vector in the sample which weakly dominates it in all coordinates. Let $p_{k,n}$ be the probability that the first vector is a maximum. The main result of the present paper is that if $k\equiv k_n$ is growing at a slower (faster) rate than a certain factor of $\log(n)$, then $p_{k,n} \rightarrow 0$ (resp. $p_{k,n}\rightarrow1$) as $n\to\infty$. Furthermore, the factor is fully characterized as a functional of $F$. We also study the effect of $F$ on $p_{k,n}$, showing that while $p_{k,n}$ may be highly affected by the choice of $F$, the phase transition is the same for all distribution functions up to a constant factor.