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.

翻译：考虑$n$个独立同分布实值随机向量，每个向量维度为$k$，其坐标独立同分布且服从一般分布函数$F$。一个向量被称为最大值当且仅当样本中不存在其他向量在所有坐标上弱支配它。设$p_{k,n}$为第一个向量成为最大值的概率。本文的主要结果是：若$k\equiv k_n$的增长速率慢于（快于）$\log(n)$的某个特定因子，则当$n\to\infty$时，$p_{k,n}\rightarrow 0$（相应地$p_{k,n}\rightarrow 1$）。此外，该因子被完全刻画为$F$的泛函。我们同时研究了$F$对$p_{k,n}$的影响，表明尽管$p_{k,n}$可能因$F$的选择而产生显著差异，但所有分布函数的相变在常数因子意义下保持一致。