## Advanced Studies in Pure Mathematics

### Limiting processes with dependent increments for measures on symmetric group of permutations

#### Abstract

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.

#### Article information

Dates
Received: 12 January 2006
Revised: 26 June 2006
First available in Project Euclid: 27 January 2019

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1548550892

Digital Object Identifier
doi:10.2969/aspm/04910041

Mathematical Reviews number (MathSciNet)
MR2405597

Zentralblatt MATH identifier
1145.60007

#### Citation

Babu, Gutti Jogesh; Manstavičius, Eugenijus; Zacharovas, Vytas. Limiting processes with dependent increments for measures on symmetric group of permutations. Probability and Number Theory — Kanazawa 2005, 41--67, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04910041. https://projecteuclid.org/euclid.aspm/1548550892