On the characteristic functional of a doubly stochastic Poisson process: Application to a narrow-band process

The characteristic functional (c.fl.) of a doubly stochastic Poisson process (DSPP) is studied and it provides us the finite dimensional distributions of the process and so its moments. It is also studied the case of a DSPP which intensity is a narrow-band process. The Karhunen–Loève expansion of its intensity is used to obtain the probability distribution function and a decomposition of this Poisson process. The covariance derived from the general c.fl. is applied in this particular DSPP.