The tail process revisited

Abstract

The tail measure of a regularly varying stationary time series has been recently introduced. It is used in this contribution to reconsider certain properties of the tail process and establish new ones. A new formulation of the time change formula is used to establish identities, some of which were indirectly known and some of which are new.

On extremal index of max-stable stationary pro­cesses

“…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).…”

“…Proof of Lemma 3.4: We give first another useful form of (2.6) proved initially in [37] and also stated for rf's in [2]. Namely, for any measurable functional…”

On aggregation of multitype Galton–Watson branching processes with immigration

“…Since the distribution of the forward tail process determines the distribution of the (whole) tail process (see Basrak and Segers [6, Theorem 3.1 (ii)]), it follows that (Y ℓ ) ℓ∈Z represents the tail process of (X ℓ ) ℓ∈Z . If m ξ = 0, then one can easily check that [20] is satisfied.…”

“…Moreover, there exists a unique measure ν α on R Z endowed with the cylindrical σ-algebra Planinić and Soulier [20]. The measure ν α is called the tail measure of (X ℓ ) ℓ∈Z .…”

“…Consequently, we may apply Theorem 3.1 in Planinić and Soulier [20] for the nonnegative measurable function H : R Z → R + given by H(y) = f • p 0,k , where f : R k+1 → R + is a measurable function with f (0) = 0. By (3.2) in Planinić and Soulier [20], we obtain…”