Regular variation in the mean and stable limits for Poisson shot noise

Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. We study the limiting behaviour of Poisson shot noise when the limits are infinite-variance stable processes. In this context a sufficient condition for this convergence turns up which is closely related to multivariate regular variation - we call it regular variation in the mean. We also show that the latter condition is necessary and sufficient for the weak convergence of the point processes constructed from the normalized noise sequence and also for the weak convergence of its extremes.