Size-biased permutation of a finite sequence with independent and identically distributed terms

TL;DR

This paper investigates the size-biased permutation of finite sequences of i.i.d. positive variables, providing new distributional results and simpler proofs, including an asymptotic description of the last few terms using Poisson coupling.

Contribution

It introduces novel asymptotic results for size-biased permutations of i.i.d. sequences, connecting them with Poisson processes and simplifying existing proofs.

Findings

Asymptotic distribution of the last few terms characterized

Poisson coupling with smallest order statistics established

New distributional properties derived for size-biased permutations

Abstract

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.