A class of weighted Poisson processes

Let N have a Poisson distribution with parameter lambda > 0, and let U-1, U-2, ... be a sequence of independent standard uniform variables, independent of N. Then the random sum N(t)= Sigma(N)(j=1) I-[0,I-t](U-j), where I-A is an indicator of the set A, is a Poisson process on [0, 1]. Replacing N by its weighted version N 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) are related to the correlation of the process increments, and are equivalent to the analogous properties of N-w. Theoretical results are illustrated through examples, which include processes with geometric and negative binomial marginal distributions. (C) 2008 Elsevier B.V. All rights reserved.