The general cumulants for a filtered point process

Abstract

This paper extends the theory of a filtered Poisson process proposed by Snyder [Random Point Processes, Wiley New York, 1975]. The cumulants for the filtered Poisson process have been given by Snyder. The filtered Poisson process is a particular form of a filtered point process in which the point process is a compound Poisson process. In practice, the point process is not always Poissionian and it might be represented by the binomial or negative binomial distribution. Thus, it is advantageous to construct the statistical properties of a filtered point process on the basis that the occurrence counting process is of both the binomial and the negative binomial types. This paper derives the characteristic functional for a filtered point process where the point process is of both the binomial and the negative binomial types. The first four cumulants for these types are also deduced. From these cumulants, we can readily obtain the basic statistics (mean, variance, coefficient of skewness, coefficient of kurtosis, and correlation coefficient) of a random variable that can be modeled as a filtered point process.