Open Access
November 2015 Size-biased permutation of a finite sequence with independent and identically distributed terms
Jim Pitman, Ngoc M. Tran
Bernoulli 21(4): 2484-2512 (November 2015). DOI: 10.3150/14-BEJ652


This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman–Pitman–Yor (PPY) [ Probab. Theory Related Fields 92 (1992) 21–39], as well as a special form of induced order statistics [ Bull. Inst. Internat. Statist. 45 (1973) 295–300; Ann. Statist. 2 (1974) 1034–1039]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simpler proofs of existing ones. Our main contribution, Theorem 25 in Section 6, describes the asymptotic distribution of the last few terms in a finite i.i.d. size-biased permutation via a Poisson coupling with its few smallest order statistics.


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Jim Pitman. Ngoc M. Tran. "Size-biased permutation of a finite sequence with independent and identically distributed terms." Bernoulli 21 (4) 2484 - 2512, November 2015.


Received: 1 October 2012; Revised: 1 March 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1362.60036
MathSciNet: MR3378475
Digital Object Identifier: 10.3150/14-BEJ652

Keywords: induced order statistics , Kingman paint box , Poisson–Dirichlet , size-biased permutation , subordinator

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
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