AISM 54, 607-620

© 2002 ISM

## Limit processes with independent increments for the Ewens sampling formula

### Gutti Jogesh Babu^{1} and Eugenijus Manstavicius^{2}

^{1}Department of Statistics, The Pennsylvania State University, 326 Thomas Building, University Park, PA 16802-2111, U.S.A.

^{2}Department of Mathematics, Vilnius University, Naugarduko str. 24, LT2006 Vilnius, Lithuania

(Received June 16, 2000; revised February 16, 2001)

Abstract.
The *Ewens sampling formula* in population
genetics can be viewed as a probability measure on the group of
permutations of a finite set of integers. Functional limit theory for
processes defined through partial sums of dependent variables with
respect to the Ewens sampling formula is developed. Techniques from
probabilistic number theory are used to establish necessary and
sufficient conditions for weak convergence of the associated dependent
process to a process with independent increments. Not many results on
the necessity part are known in the literature.

Key words and phrases:
Random partitions, cycle, population genetics, permutations, law of large numbers, probabilistic number theory, Skorohod topology, slowly varying function, functional limit theorem.

**Source**
(TeX , DVI )