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 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 pot 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.