Regular point processes and their detection

A class of point processes that possess intensity functions are studied. The processes of this class, which seem to include most point processes of practical interest, are called regular point processes (RPP's). Expressions for the evolution of these processes and especially for their joint occurrence statistics are derived. Compound RPP's, which are RPP's whose intensity functions are themselves stochastic processes, are shown to be RPP's whose intensity functions are given as the causal minimum mean-squared-error (MMSE) estimates of the given intensity functions. The superposition of two independent RPP's is shown to yield an RPP whose intensity is given as a causal least squares estimate of the appropriate combination of the two given intensity functions. A general likelihood-ratio formula for the detection of compound RPP's is obtained. Singular detection cases are characterized. Detection procedures thai use only the total number of counts are discussed. As an example, the optimal detection scheme for signals of the random-telegraph type with unknown transition intensities is derived.