Distribution of the mean value for certain random measures

Abstract

Let Τ be a probability measure on [0,1]. We consider a generalization of the classic Dirichlet process — the random probability measure $$F = \sum {P_i \delta _{X_i } } $$ , where X={Xi} is a sequence of independent random variables with the common distribution Τ and P={Pi} 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 Τ.