Role of the doubly stochastic Neyman type-A and Thomas counting distributions in photon detection

The Neyman type-A and Thomas counting distributions provide a useful description for a broad variety of phenomena from the distribution of larvas on small plots of land to the distribution of galaxies in space. They turn out to provide a good description for the counting of photons generated by multiplied Poisson processes, as long as the time course of the multiplication is short compared with the counting time. Analytic expressions are presented for the probability distributions, moment generating functions, moments, and variance-to-mean ratios. Sums of Neyman type-A and Thomas random variables are shown to retain their form under the constraint of constant multiplication parameter. Conditions under which the Neyman type-A and Thomas converge in distribution to the fixed multiplicative Poisson and to the Gaussian are presented. This latter result is most important for it provides a ready solution to likelihood-ratio detection, estimation, and discrimination problems in the presence of many kinds of signal and noise. The doubly stochastic Neyman type-A, Thomas, and fixed multiplicative Poisson distributions are also considered. A number of explicit applications are presented. These include (1) the photon counting scintillation detection of nuclear particles, when the particle flux is low, (2) the photon counting detection of weak optical signals in the presence of ionizing radiation, (3) the design of a star-scanner spacecraft guidance system for the hostile environment of space, (4) the neural pulse counting distribution in the cat retinal ganglion cell at low light levels, and (5) the transfer of visual signal to the cortex in a classical psychophysics experiment. A number of more complex contagious distributions arising from multiplicative processes are also discussed, with particular emphasis on photon counting and direct-detection optical communications.