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.
Categories: math.PR