A class of weighted Poisson processes

Let N have a Poisson distribution with parameter λ>0, and let U1,U2,… be a sequence of independent standard uniform variables, independent of N. Then the random sum N(t)=∑j=1NI[0,t](Uj), where IA is an indicator of the set A, is a Poisson process on [0,1]. Replacing N by its weighted version Nw, we obtain another process with weighted Poisson marginal distributions. We then derive the basic properties of such processes, which include marginal …