Extremes of Moving Averages of Stable Processes

In this paper we study extremes of non-normal stable moving average processes, i.e., of stochastic processes of the form X(t) = Σ a(λ - t)Z(λ) or X(t) = ∫ a(λ - t) dZ(λ), where Z(λ) is stable with index α < 2. The extremes are described as a marked point process, consisting of the point process of (separated) exceedances of a level together with marks associated with the points, a mark being the normalized sample path of $X(t)$ around an exceedance. It is proved that this marked point process converges in distribution as the level increases to infinity. The limiting distribution is that of a Poisson process with independent marks which have random heights but otherwise are deterministic. As a byproduct of the proof for the continuous-time case, a result on sample path continuity of stable processes is obtained.