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Extremal behavior of regularly varying stochastic processes. (English) Zbl 1070.60046
Summary: We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the continuous mapping theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the continuous mapping theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the continuous mapping theorem we derive the tail behavior of filtered regularly varying Lévy processes.
MSC:
60G70Extreme value theory; extremal processes (probability theory)
60F17Functional limit theorems; invariance principles
60G17Sample path properties
60G07General theory of stochastic processes