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Extremes of vector-valued locally additive Gaussian fields with application to double crossing probabilities

Ievlev Pavel, Kriukov Nikolai

TL;DR

The paper develops a comprehensive framework to obtain exact high-exceedance asymptotics for vector-valued Gaussian fields with a simple additive local structure near the most likely exceedance point. By extending the double-sum method to non-homogeneous, vector-valued fields and leveraging a suite of tools (Borell–Tsirelson–Piterbarg inequalities, Local Pickands lemmas, and quadratic programming), the authors derive an explicit main asymptotic form that factors into a Pickands-type constant, a multi-dimensional generalized-variance term, and the one-dimensional marginal exceedance probability. The results are then applied to double crossing problems, i.e., the probability that a process crosses a high positive barrier and a low negative barrier within a finite horizon, including stationary and fractional Brownian motion cases, with detailed regimes determined by the underlying smoothness indices. These contributions generalize prior work by Debicki et al. (2019) to non-homogeneous vector fields and provide practically computable constants for applications in risk, queueing, and related areas. Overall, the work advances the theory of high-level excursions for multivariate Gaussian fields and offers precise asymptotics for complex crossing events.

Abstract

The asymptotic analysis of high exceedance probabilities for Gaussian processes and fields has been a blooming research area since J. Pickands introduced the now-standard techniques in the late 60's. The \textit{vector-valued} processes, however, have long remained out of reach due to the lack of some key tools including Slepian's lemma, Borell-TIS and Piterbarg inequalities. In a 2020 paper by K. Debicki, E. Hashorva and L. Wang, the authors extended the double-sum method to a large class of vector-valued processes, both stationary and non-stationary. In this contribution we make one step forward, extending these results to a simple yet rich class of non-homogenous vector-valued Gaussian \textit{fields}. As an application of our findings, we present an exact asymptotic result for the probability that a real-valued process first hits a high positive barrier and then a low negative barrier within a finite time horizon.

Extremes of vector-valued locally additive Gaussian fields with application to double crossing probabilities

TL;DR

The paper develops a comprehensive framework to obtain exact high-exceedance asymptotics for vector-valued Gaussian fields with a simple additive local structure near the most likely exceedance point. By extending the double-sum method to non-homogeneous, vector-valued fields and leveraging a suite of tools (Borell–Tsirelson–Piterbarg inequalities, Local Pickands lemmas, and quadratic programming), the authors derive an explicit main asymptotic form that factors into a Pickands-type constant, a multi-dimensional generalized-variance term, and the one-dimensional marginal exceedance probability. The results are then applied to double crossing problems, i.e., the probability that a process crosses a high positive barrier and a low negative barrier within a finite horizon, including stationary and fractional Brownian motion cases, with detailed regimes determined by the underlying smoothness indices. These contributions generalize prior work by Debicki et al. (2019) to non-homogeneous vector fields and provide practically computable constants for applications in risk, queueing, and related areas. Overall, the work advances the theory of high-level excursions for multivariate Gaussian fields and offers precise asymptotics for complex crossing events.

Abstract

The asymptotic analysis of high exceedance probabilities for Gaussian processes and fields has been a blooming research area since J. Pickands introduced the now-standard techniques in the late 60's. The \textit{vector-valued} processes, however, have long remained out of reach due to the lack of some key tools including Slepian's lemma, Borell-TIS and Piterbarg inequalities. In a 2020 paper by K. Debicki, E. Hashorva and L. Wang, the authors extended the double-sum method to a large class of vector-valued processes, both stationary and non-stationary. In this contribution we make one step forward, extending these results to a simple yet rich class of non-homogenous vector-valued Gaussian \textit{fields}. As an application of our findings, we present an exact asymptotic result for the probability that a real-valued process first hits a high positive barrier and then a low negative barrier within a finite time horizon.
Paper Structure (41 sections, 25 theorems, 426 equations)

This paper contains 41 sections, 25 theorems, 426 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Constants
  4. Main theorem
  5. Double crossing probabilities: general facts
  6. Double crossing probabilities: stationary case
  7. Double crossing probabilities: fBm case
  8. Auxiliary results
  9. Quadratic programming problem
  10. Borell-TIS and Piterbarg inequalities
  11. Local Pickands lemma
  12. Integral estimate
  13. Proof of the main theorem
  14. Log-layer bound
  15. Double sum bound
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\bm{X} ( \bm{t} )$, $\bm{t} \in [ \bm{0}, \bm{T} ] \subset \mathbb{R}^n$ be a centered $\mathbb{R}^d$-valued Gaussian random field, satisfying Assumptions A1, A2 and A3. Then with $G$ defined by def:G, $( C_{i, k} )_{i, k \in ( \mathcal{J} \cup \mathcal{K} ) \cap \mathcal{F}}$ any family of matrices satisfying A2.6-D-as-sum, and

Theorems & Definitions (54)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Corollary 1
  • Remark 5
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • ...and 44 more