The tail process and tail measure of continuous time regularly varying stochastic processes

Soulier, Philippe

doi:10.1007/s10687-021-00417-3

Cited by 14 publications

(55 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof of Corollary 2.1. First, let us notice that H δ Z = 0 for some δ ≥ 0 if and only if S δ (Z) = ∞ almost surely, which is a direct implication of [14], [22], and [34,35,36] ; this has already been discussed in [17], [9], and [37] for the case d = 1, and [20] for the d-dimensional discrete setup. Note in passing that in [14]…”

Section: Proofsmentioning

confidence: 80%

See 1 more Smart Citation

On the continuity of Pickands constants

2022

View full text Add to dashboard Cite

For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} <\infty$ for $\delta \ge 0$ , where $0 \mathbb{Z}^d\,:\!=\,\mathbb{R}^d$ , and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for Pickands constants $H_{Z}^\delta$ , defined for $Z(t)= \exp( \sqrt{ 2} B_\alpha (t)- \lvert {t} \rvert^{2\alpha })$ , $t\in \mathbb{R}$ , with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$ . The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants.

show abstract

Section: Proofsmentioning

confidence: 80%

“…The two expressions in (1.2) paved the way for simulations of Pickands constants and inspired new formulas for extremal indices of stationary time series; see e.g. [12], [19], [11], [8], [30], [20], and [37].…”

Section: Introductionmentioning

confidence: 99%

On the continuity of Pickands constants

2022

View full text Add to dashboard Cite

show abstract

“…which together with (4.3) yields lim δ↓0 H δ Z = H 0 Z . Proof of Corollary 1 First, let us notice that H δ Z = 0 for some δ ≥ 0 if and only if S δ (Z) = ∞ almost surely, which is a direct implication of [15,21] and [33][34][35]; this has been already discussed in [10,17,36] for the case d = 1 and [20] for the d-dimensional discrete setup. Note in passing that in [15] Z is such that P{sup t∈R d Z(t) > 0} = 1.…”

Section: Proofsmentioning

confidence: 75%

“…2)], see also [36] for other equivalent formulations. We note in passing that (2.1) is initially derived for Z…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the continuity of Pickands constants

Dȩbicki¹,

Hashorva²,

Michna³

2021

Preprint

View full text Add to dashboard Cite

For a non-negative separable random field Z(t), t ∈ R d satisfying some mild assumptions we show thatfor δ ≥ 0 where 0Z d := R d and prove that H 0 Z can be approximated by H δ Z if δ tends to 0. These results extend the classical findings for the Pickands constants H δ Z , defined for Z(t) = exp √ 2Bα(t) − |t| 2α , t ∈ R with Bα a standard fractional Brownian motion with Hurst parameter α ∈ (0, 1]. The continuity of H δ Z at δ = 0 is additionally shown for two particular extensions of Pickands constants.

show abstract

Point Process Convergence for Regularly Varying Sequences

Mikosch,

Wintenberger

2024

Springer Series in Operations Research and Financial Engineering

View full text Add to dashboard Cite

The tail process and tail measure of continuous time regularly varying stochastic processes

Cited by 14 publications

References 44 publications

On the continuity of Pickands constants

On the continuity of Pickands constants

On the continuity of Pickands constants

Point Process Convergence for Regularly Varying Sequences

Contact Info

Product

Resources

About