  This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:=
X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le
X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the
positive integers $(P_1, P_2, \ldots)$ governed by a residual allocation model
(also called a Bernoulli sieve) $P_j:= H_j \prod_{i=1}^{j-1}(1-H_i)$ for a
sequence of independent random hazard variables $H_i$ which are identically
distributed according to some distribution of $H \in (0,1)$ such that $- \log(1
- H)$ has a non-lattice distribution with finite mean $\mu_{\mbox{log}}$. As
$n\to \infty$ the finite dimensional distributions of the gaps
$\widehat{G}_{i:n}$ converge to those of limiting gaps $G_i$ which are the
numbers of points in a stationary renewal process with i.i.d. spacings $-
\log(1 - H_j)$ between times $T_{i-1}$ and $T_i$ of births in a Yule process,
that is $T_i := \sum_{k=1}^i \varepsilon_{k}/k$ for a sequence of i.i.d.
exponential variables $\varepsilon_k$ with mean 1. A consequence is that the
mean of $\widehat{G}_{i:n}$ converges to the mean of $G_i$, which is $1/(i
\mu_{\mbox{log}} )$. This limit theorem simplifies and extends a result of
Gnedin, Iksanov and Roesler for the Bernoulli sieve.
