Electronic Journal of Probability

Extremes and gaps in sampling from a GEM random discrete distribution

Jim Pitman and Yuri Yakubovich

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Abstract

We show that in a sample of size $n$ from a $\mathsf{GEM} (0,\theta )$ 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 $\mathsf{GEM} (0,\theta )$ sampling statistics, including the Ewens and Donnelly–Tavaré 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.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 44, 26 pp.

Dates
Received: 10 February 2017
Accepted: 21 April 2017
First available in Project Euclid: 3 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1493777020

Digital Object Identifier
doi:10.1214/17-EJP59

Mathematical Reviews number (MathSciNet)
MR3646070

Zentralblatt MATH identifier
1364.60069

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60G09: Exchangeability 60F05: Central limit and other weak theorems

Keywords
GEM samples order statistics sample maximum random discrete distribution size-biased permutation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pitman, Jim; Yakubovich, Yuri. Extremes and gaps in sampling from a GEM random discrete distribution. Electron. J. Probab. 22 (2017), paper no. 44, 26 pp. doi:10.1214/17-EJP59. https://projecteuclid.org/euclid.ejp/1493777020


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