Extremes for shot noise processes with heavy tailed amplitudes

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, X(t) = ∑τk≦t Akh(t – τk), where {τk} are the points of a renewal process and {Ak} are independent, identically distributed with density function having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0,1) is possible. Two-dimensional point processes of the form ∑j δj/n,u^←n(X(τj)) are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.