Distribution of the mean value for certain random measures

Let be a probability measure on 0,1]. We consider a generalization of the classic Dirichlet process — the random probability measure , where X={X_i} is a sequence of independent random variables with the common distribution and P={P_i} is independent of X and has the two-parameter Poisson-Dirichlet distribution PD(, ) on the unit simplex. The main result is the formula connecting the distribution of the random mean value xdF(x) with the parameter measure .Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 268–279.Supported by the Russian Foundation for Basic Research, grant 96-01-00676.