The prefix exchange distance of a permutation is the length its shortest factorisation into transpositions that all contain 1. Using probabilistic approach, we obtain expressions for mean and variance, prove asymptotic normality distribution this random verifying Ewens sampling formula. Analogous results in uniform setting follow as simple corollaries.

The clustering of agents in the market is a typical problem dealt with by the new approaches to macroeconomic modeling, that describe macroscopic variables in terms of the behavior of a large collection of microeconomic entities. Clustering has a lot of economical interpretations [3], that are often described by Ewens Sampling Formula (ESF). This formula can be traced back to Fisher as “species...

Statistical tests of the neutrality hypothesis that are based on the sampling theory of Ewens (1972) require the assumption of panmixia. It is proposed that for a population comprising numerous local populations with weak gene flow among them, tests based on Ewens' theory can be applied separately to samples from each local population. At low levels of gene flow, migration acts primarily like m...

In this paper, we establish the cutoff phenomena for the discrete time infinite alleles Moran model. If M is the population size and μ is the mutation rate, we find a cutoff time of log(Mμ)/μ generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generat...

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...

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. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a p...