Tail measure and spectral tail process of regularly varying time series

Abstract

The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main result is to prove in an abstract framework that there is a one to one correspondance between these two objets, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For non negative time series, we recover results explicitly or implicitly known in the theory of maxstable processes. Tail measures on a metric space.2.1. Framework. The mathematical setting is the following. Let pE, Eq be a measurable cone, that is a measurable space together with a multiplication by positive scalars pu, xq P p0, 8qˆE Þ Ñ ux P E , which is measurable with respect to the product σ-field Bp0, 8q b E{E and satisfies 1x " x , upvpxqq " puvqx , u, v ą 0 , x P E .We assume that the cone admits a zero element 0 E P E such that u0 E " 0 E for all u ą 0 and that it is endowed with a pseudonorm, i.e. a measurable function }¨} E : E Þ Ñ r0, 8q such that }ux} E " u}x} E for all u ą 0, x P E and }x} E " 0 implies x " 0 E . The triangle inequality is not required.The space E Z of E-valued sequences is endowed with the cylinder σ-algebra F " E bZ and a generic sequence is denoted x " px h q hPZ . The sequence identically equal to 0 E is denoted by imsart-aap ver.

On extremal index of max-stable stationary pro­cesses

“…valid again for all measurable functionals F as above, see e.g., [2,19]. We note in passing that with the same arguments as in [19] it can be shown that (2.6) is equivalent to the so-called time-change formula proven in [2] for multivariate regularly varying rf's.…”

“…valid again for all measurable functionals F as above, see e.g., [2,19]. We note in passing that with the same arguments as in [19] it can be shown that (2.6) is equivalent to the so-called time-change formula proven in [2] for multivariate regularly varying rf's. Next, since for stationary X we have that (2.2) holds, then in view of [2,19] X is a multivariate regularly varying rf and Y is the so-called tail rf of X, whereas Θ is the so-called spectral tail rf.…”

“…Without loss of generality, we shall focus on the class of max-stable rf's with Fréchet marginals. Since these are limiting rf's, see e.g., [19], our formulas for their extremal indices are valid (with obvious modifications) also for the candidate extremal index of more general stationary regularly varying rf's (see [37] for recent findings). Studying max-stable rf's, instead of these more general rf's is also justified by Theorem 2.3 stated in Section 2 and Remark 2.4 iii).…”

“…A1: Z(t) → 0 almost surely as t → ∞; A2: Θ(t) → 0 almost surely as t → ∞. The equivalence of A1 and A2 is known, see e.g., [19]. For calculation of θ X , typically in the literature A1 or A2 is assumed, see e.g., [27].…”

“…It is well-known that a max-stable rf X with Fréchet marginals is a multivariate regularly varying rf. For general multivariate regularly varying rf's which are not max-stable, there is no spectral process Z as in our case of max-stable X and therefore the rf's Θ h , h ∈ Z d are defined via a conditional limit, see e.g., [19,41] and (2.1) below. The key advantage in the framework of max-stable rf's is that Θ h is directly obtained by tilting a given spectral rf Z and thus a limiting argument can be avoided if Z is known.…”