The general cumulants for a filtered point process

This paper extends the theory of a filtered Poisson process proposed by Sny der [Random Point Processes, Wiley New York, 1975]. The cumulants for the f iltered Poisson process have been given by Snyder. The filtered Poisson pro cess is a particular form of a filtered point process in which the point pr ocess 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 statistica l properties of a filtered point process on the basis that the occurrence c ounting process is of both the binomial and the negative binomial types. Th is 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 thes e cumulants, we can readily obtain the basic statistics (mean, variance, co efficient of skewness, coefficient of kurtosis, and correlation coefficient ) of a random variable that can be modeled as a filtered point process. (C) 2001 Elsevier Science Inc. All rights reserved.