Power-law shot noise and its relationship to long-memory /spl alpha/-stable processes

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