## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probability and Number Theory — Kanazawa 2005, S. Akiyama, K. Matsumoto, L. Murata and H. Sugita, eds. (Tokyo: Mathematical Society of Japan, 2007), 41 - 67

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

Gutti Jogesh Babu, Eugenijus Manstavičius, and Vytas Zacharovas

#### 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

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60F17: Functional limit theorems; invariance principles 11K65: Arithmetic functions [See also 11Nxx]

**Keywords**

Cycle Ewens sampling formula functional limit theorem random partitions tightness slowly varying function weak convergence

#### 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