  We show that in a sample of size $n$ from a GEM$(0,\theta)$ random discrete
distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order
statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention
$G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an
infinite sequence of independent geometric$(i/(i+\theta))$ variables $G_i$.
This extends a known result for the minimum $X_{1:n}$ to other gaps in the
range of the sample, and implies that the maximum $X_{n:n}$ has the
distribution of $1 + \sum_{i=1}^n G_i$, hence the known result that $X_{n:n}$
grows like $\theta\log(n)$ as $n\to\infty$, with an asymptotically normal
distribution. Other consequences include most known formulas for the exact
distributions of GEM$(0,\theta)$ sampling statistics, including the Ewens and
Donnelly--Tavar\'e sampling formulas. For the two-parameter
GEM$(\alpha,\theta)$ distribution we show that the maximal value grows like a
random multiple of $n^{\alpha/(1-\alpha)}$ and find the limit distribution of
the multiplier.
