On the representation of a continuous stochastic intensity by Poisson shot noise

In many applications a Poisson shot noise (PSN) process is said to statistically "represent" its intensity process. In this paper an investigation is made of the relationship between a PSN process and its intensity, when the latter is a sample function of a continuous stochastic process. The difference of the moments and the mean-square difference between the two processes are examined. The continuity assumption on the intensity permits the development of a sequence of moment relationships in which the effect of the PSN parameters can be seen. The results simplify and afford some degree of physical interpretation when the component functions of the PSN are "rectangular," or when the intensity process does not vary appreciably over their time width. An integral equation is derived that defines the component function that minimizes the mean-square difference between the two processes. It is shown that a "degenerate" form of component function induces complete statistical equality of the two processes. The problem has application to optical communication systems using photodetectors.