A class of weighted Poisson processes

Let NN have a Poisson distribution with parameter λ>0λ>0, and let U1,U2,…U1,U2,… be a sequence of independent standard uniform variables, independent of NN. Then the random sum N(t)=∑j=1NI[0,t](Uj), where IAIA is an indicator of the set AA, is a Poisson process on [0,1][0,1]. Replacing NN by its weighted version NwNw, we obtain another process with weighted Poisson marginal distributions. We then derive the basic properties of such processes, which include marginal and joint distributions, stationarity of the increments, moments, and the covariance function. In particular, we show that properties of overdispersion and underdispersion of N(t)N(t) are related to the correlation of the process increments, and are equivalent to the analogous properties of NwNw. Theoretical results are illustrated through examples, which include processes with geometric and negative binomial marginal distributions.