Means of nonparametric priors based on increasing additive processes

We provide a survey on some distributional results concerning means of random probability measure constructed via suitable transformation of an increasing additive process (abbreviated as IAP), i.e. increasing, not necessarily hopmogeneous, and purely discontinuous Lévy process. In particular, we deal with normalized random measures having independent increments and with neutral to the right (NTR) random probability measures. The former are obtained by normalizing IAPs and the exact distribution of a mean is found by resorting to a well-known inversion formula for characteristic functions. Morevoer, expressions of the posterior distributions of those means, in the presence of exchangeable observations, are given. Also the latter may be characterized in terms of IAPs and we show the connection between a mean of a NTR prior and the so-called exponential functional. We study finiteness and absolute continuity of these functionals and provide some formulae for computing their moments, provided they exist. All the results contained in the first section can be found in Lijoi, Regazzini and Pruenster (2000) whereas those of Section 2 are based on Epifani, Lijoi and Pruenster (2002).