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VOL. 49 | 2007 Limiting processes with dependent increments for measures on symmetric group of permutations
Gutti Jogesh Babu, Eugenijus Manstavičius, Vytas Zacharovas

Editor(s) Shigeki Akiyama, Kohji Matsumoto, Leo Murata, Hiroshi Sugita


A family of measures on the set of permutations of the first $n$ integers, known as Ewens sampling formula, arises in population genetics. In a series of papers, the first two authors have developed necessary and sufficient conditions for the weak convergence of a partial sum process based on these measures to a process with independent increments. Under very general conditions, it has been shown that a partial sum process converges weakly in a function space if and only if a related process defined through sums of independent random variables converges. In this paper, a functional limit theory is developed where the limiting processes need not be processes with independent increments. Thus, under Ewens sampling formula, the limiting process of the partial sums of dependent variables differs from that of the associated process defined through the partial sums of independent random variables.


Published: 1 January 2007
First available in Project Euclid: 27 January 2019

zbMATH: 1145.60007
MathSciNet: MR2405597

Digital Object Identifier: 10.2969/aspm/04910041

Primary: 60C05
Secondary: 11K65 , 60F17

Keywords: cycle , Ewens sampling formula , Functional limit theorem , Random partitions , slowly varying function , tightness , weak convergence

Rights: Copyright © 2007 Mathematical Society of Japan


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