Poisson Process and Shot Noise

Abstract

While the Wiener process has continuous sample paths and continuously changes by small increments, the Poisson process is integer valued with discontinuous sample paths and changes infrequently by unit increments. So, it is well adapted to model discrete random phenomena, such as photon counts in optics, the number of tasks processed by a computer system, or random spikes in a neural pathway. Like the Wiener process, the Poisson process has independent increments and can be constructed in several equivalent ways, which are described in Sect. 7.2. The times at which the Poisson process undergoes jumps are called epochs, and the interarrival times (the time between two epochs) are exponentially distributed. In this context, as explained in Sect. 7.3, the memoryless property of exponential distributions has an interesting consequence concerning the distribution of the residual time until the occurrence of the next epoch after an arbitrary reference time. As shown in Sect. 7.5, Poisson processes have also the the interesting feature that they remain Poisson after either merging, or splitting randomly into two lower rate processes. Finally, Sect. 7.6 examines the statistical properties of shot noise, which is a form of noise occurring in electronic devices, such as vacuum tubes or optical detectors, when electrons or photons arriving at random instants (modeled typically by the epochs of a Poisson process) trigger an electronic circuit response.