  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.
