Extremes of vector-valued locally additive Gaussian fields with application to double crossing probabilities
Abstract
We derive exact asymptotics for high exceedance probabilities of vector-valued Gaussian fields that are locally additive near the point of maximal variance. As an application, we obtain the double crossing probability for stationary processes and fractional Brownian motion.
Keywords
Vector-valued Gaussian fields,
Extremes,
Double crossing probability,
Fractional Brownian motion,
Pickands constant
Overview
We study high exceedance probabilities
$$ \mathbb{P}\{\exists\, t \in [0, T]: \boldsymbol{X}(t) > u\boldsymbol{b}\} \quad \text{as} \quad u \to \infty $$for centered $\mathbb{R}^d$-valued Gaussian random fields. The main contribution is extending the double-sum method to non-homogeneous vector-valued Gaussian fields that are “locally additive” near high exceedance points.
Main Results
- Theorem 1: Exact asymptotic equivalence for exceedance probabilities involving Pickands-type and geometric constants
-
Double Crossing Probability: Asymptotics for
$$ \mathbb{P}\{\exists \, t, s: X(t) > au, X(s) < -bu\} $$
Applications
- Stationary Processes: Different regimes depending on smoothness parameter $\alpha$ (Pickands-type for $\alpha < 1$, Piterbarg-type for $\alpha = 1$)
- Fractional Brownian Motion: Results for all Hurst indices $H \in (0,1)$ with regime changes at $H = 1/2$
BibTeX
@article {MR4865024,
AUTHOR = {Ievlev, Pavel and Kriukov, Nikolai},
TITLE = {Extremes of vector-valued locally additive {G}aussian fields
with application to double crossing probabilities},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {30},
YEAR = {2025},
PAGES = {Paper No. 26, 43},
MRCLASS = {60G15 (60G60 60G70)},
MRNUMBER = {4865024},
DOI = {10.1214/25-ejp1288},
URL = {https://doi.org/10.1214/25-ejp1288}
}