Extremal behavior of regularly varying stochastic processes

Abstract

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 Le´vy processes. r 2004 Elsevier B.V. All rights reserved. MSC: primary 60F17; 60G17; secondary 60G07; 60G70

Regular variation for measures on metric spaces

“…For instance, on the Skorohod space D[0, 1], regular variation is formulated using polar coordinates and a notion of convergence for boundedly finite measures (see [13] and [12]). Also in this case the original space is changed by introducing "points at infinity" in order to turn sets bounded away from 0 in the original space into metrically bounded sets.…”

High-level dependence in time series models

“…(i) Regular variation of stochastic processes has been defined in Hult and Lindskog (2005). Our definition concerns only the finite-dimensional distributions.…”

“…Our definition concerns only the finite-dimensional distributions. Consequently, it is for continuous-time processes weaker than the definition presented in Hult and Lindskog (2005), which requires an additional tightness condition. (ii) The equivalence of Definitions (a) and (b) above is based on the following transformation to polar coordinates.…”

Extremes of independent stochastic processes: a point process approach

“…The reader should refer to Appendix 2.6 in Daley and Vere-Jones [3] or to section 2 of Davis and Mikosch [4]. This has also closed connections with the theory of regular variations, see Hult and Lindskog [11,12].…”

“…In the terminology of Hult and Lindskog [11,12], the condition i) in Proposition 2.1 means that the sequence X n is regularly varying. A particularly important case is when E is endowed with a strucure of cone, i.e.…”