We consider a class of random discrete distributions which can be constructed by means of a recursive and branching splitting of unity. Random discrete distributions are in one-to-one correspondence with the partition structures in the sense of Kingman. We investigate the number of blocks in an n-element sample from the random exchangeable partition of N. It turns out that this number behaves as n α times some random factor, for some α between 0 and 1. The proof involves an application of known results on a general Crump--Mode--Jagers branching process.
This is a joint work with Alexander Gnedin.