Stick breaking process generated by virtual permutations with Ewens distribution

Abstract

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n $\in \mathfrak{S}_n$ such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space$\mathfrak{S}^\infty = \mathop {\lim }\limits_ \leftarrow \mathfrak{S}_n$ of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles.