Distribution of the mean value for certain random measures

Let τ be a probability measure on [0,1]. We consider a generalization of the classic Dirichlet process - the random probability measure F = Σ Pi δxi, where X = {Xi} is a sequence of independent random variables with the common distribution r and P = {P i} is independent of X and has the two-parameter Poisson-Dirichlet distribution PD(α,θ) on the unit simplex. The main result is the formula connecting the distribution μ of the random mean value ∫ xdF(x) with the parameter measure T. © 1999 Kluwer Academic/Plenum Publishers.