Distributions of the mean values for some random measures

N. V. Tsilevich

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 268-279

Résumé

Let $\tau$ 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 $X=\{X_i\}$ is a sequence of independent random variables with the common distribution $\tau$ and $P=\{P_i\}$ is independent of $X$ and has the two-parameter Poisson–Dirichlet distribution $PD(\alpha,\theta)$ on the unit simplex. The main result is the formula connecting the distribution $\mu$ of the random mean value $\int x\,dF(x)$ with the parameter measure $\tau$.