Stick breaking process generated by virtual permutations with Ewens distribution

S. V. Kerov   ; N. V. Tsilevich

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part I, Tome 223 (1995), pp. 162-180

Résumé

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 permutations $w_n\in\mathfrak S_n$ such that for all $n=1,2,\dots$, $w_{n-1}=w'_n$ 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 $\mathfrak S_n=\varprojlim\mathfrak S_n$ of virtual permutations is identified with the cube $[0,1]^\infty$. Bibliography: 8 titles.