Infinitely divisible limit processes for the Ewens sampling formula
This paper is concerned with infinitely divisible limit processes for the Ewens sampling formula. The formula states that if there are no selection effects, the numbers of alleles \(k_1,k_2,\dots,k_n\) represented \(1,2,\dots,n\) times, respectively, in a sample of \(n\) genes are \[ \frac{n!}{v(v+1)\cdots (v+n-1)}\prod^n_{j=1} (v/j)^{k_j}/k_j,\quad v> 0,\;k_j\geq 0, \] where \(k_1 + 2k_2+\cdots nk_n = n\). The authors show that under very general conditions a partial sum process of dependent variables converges weakly in a function space if and only if the corresponding process for independent random variables converges weakly. Necessary and sufficient conditions are established for weak convergence to a stable process, but it is shown by a counterexample that these conditions are not necessary for the one-dimensional convergence.
scientific article; zbMATH DE number 1912617