Power-law shot noise and its relationship to long-memory (alpha)-stable processes

We consider the shot noise process, whose associated impulse response is a decaying power-law kernel of the form t(beta)/2-1 . We show that this power-law Poisson model gives rise to a process that, at each time instant, is an (alpha)-stable random variable if (beta)<1. We show that although the process is not (alpha)-stable, pairs of its samples become jointly (alpha)-stable as the distance between them tends to infinity. It is known that for the case (beta)>1, the power-law Poisson process has a power-law spectrum. We show that, although in the case (beta)<1 the 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 as well as to synthesize a random process with long-range dependence