Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels

Authors: Pavel Ievlev, Timofei Shashkov, Svyatoslav Novikov
Published in: submitted to Stochastic Processes and their Applications (2026)
Date:
arxiv

Abstract

We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes, covering classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals and shrinking-window Parisian persistence functionals, together with conditional limit laws for the first exceedance time.

Keywords

Vector-valued Gaussian processes, Path functionals, Sojourn times, Parisian persistence, Pickands constant, Piterbarg constant

Overview

We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form

$$ \mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$

where $\boldsymbol{X}$ is a centered $\mathbb{R}^d$-valued Gaussian process and $\Gamma$ belongs to a broad class satisfying natural monotonicity, scaling, no-atom, and continuity assumptions.

Scope

The class covers classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals, and, through a local-footprint extension, shrinking-window Parisian persistence functionals.

Main Results

  • Exact asymptotics in the stationary case.
  • Exact asymptotics in the non-stationary case when the inverse generalized variance has a unique minimizer at the boundary point. The theorem covers the regimes $\alpha < \beta$, $\alpha = \beta$, and $\alpha > \beta$, which lead respectively to Pickands-type, Piterbarg-type, and deterministic limiting constants.
  • Conditional limit laws for the first exceedance time.

BibTeX

@article{ievlev_shashkov_novikov_2026,
  title={Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels},
  author={Ievlev, Pavel and Shashkov, Timofei and Novikov, Svyatoslav},
  journal={submitted to Stochastic Processes and their Applications},
  year={2026},
  eprint={2607.16502},
  archivePrefix={arXiv},
  primaryClass={math.PR}
}