LIMIT PROCESSES WITH INDEPENDENT INCREMENTS FOR THE EWENS SAMPLING FORMULA

GUTTI JOGESH BABU^{1} AND EUGENIJUS MANSTAVIČIUS^{2}

1Department of Statistics, The Pennsylvania State University, 326 Thomas Building,

University Park, PA 16802-2111, U.S.A.

2Department 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.