Power-law shot noise and its relationship to long-memory α-stable processes

We consider the shot noise process, whose associated impulse response is a decaying power-law kernel of the form . We show that this power-law Poisson model gives rise to a process that, at each time instant, is an -stable random variable, if . We show that although the process is not -stable, pairs of its samples become jointly -stable as the distance between them tends to infinity. It is known that for the case the power-law Poisson process has a power-law spectrum. We show that, although in the case , power spectrum does not exist, the process still exhibits long-memory in a generalized sense. The power-law shot noise process appears in many applications in engineering and physics. The proposed results can be used to study such processes, and also to synthesize a random process with long-range dependence. Keywords—shot noise, long-range dependence, alpha-stable, power-law, Poisson process.