[Paper Review] Extremes of Gaussian Random Fields with maximum variance attained over smooth curves

Peng Liu|arXiv (Cornell University)|Dec 22, 2016

Financial Risk and Volatility Modeling10 references3 citations

TL;DR

This paper derives exact asymptotics for the tail probability of the supremum of a two-dimensional centered Gaussian random field with maximum variance attained over a smooth curve, extending prior work on unique maximum variance points. Using generalized Piterbarg-Pickands theory, it establishes precise tail decay rates involving generalized Pickands and Piterbarg constants, with application to the extremes of aggregated fractional Brownian motions.

ABSTRACT

Let $X(s,t), (s,t)\in E$, with $E\subset \mathbb{R}^2$ a compact set, be a centered two dimensional Gaussian random field with continuous trajectories and variance function $σ(s,t)$. Denote by $\mathcal{L}=\{(s,t): σ(s,t)=\max_{(s',t')\in E}σ(s',t')\}$. In this contribution, we derive the exact asymptotics of $\mathbb{P}\left(\sup_{(s,t)\in E}X(s,t)>u ight)$, as $u o\infty$, under condition that $\mathcal{L}$ is a smooth curve. We illustrate our findings by an application concerned with extremes of the aggregation of two independent fractional Brownian motions.

Motivation & Objective

• To extend existing asymptotic theory for Gaussian random fields beyond the case of unique maximum variance points.
• To analyze the tail behavior of the supremum of a Gaussian field when the variance achieves its maximum over a smooth curve rather than a single point.
• To generalize the Piterbarg-Pickands framework to non-unique maximum variance sets with smooth structure.
• To provide exact asymptotic expressions for the tail probability of the supremum under general regular variation assumptions on correlation and variance functions.
• To apply the theoretical results to the extremes of aggregated fractional Brownian motions, a problem arising in Shepp statistics and stochastic processes with long-range dependence.

Proposed method

• The paper employs a generalization of the Piterbarg-Pickands lemma to handle non-unique maxima of the variance function over a smooth curve.
• It models the local behavior of the Gaussian field near the maximum variance curve using a bivariate Gaussian process with regularly varying correlation and variance structures.
• The method relies on the use of generalized Pickands and Piterbarg constants, defined via fractional Brownian motion and their supremum distributions.
• The asymptotic analysis is conducted via uniform expansions of tail probabilities over shrinking neighborhoods of the maximum variance curve, using a normalization based on inverse hazard functions.
• The key technical tool is a uniform convergence result (Lemma 6.1) for the tail probability of the supremum over compact sets under scaling, ensuring consistency in the limit.
• The derivation involves careful analysis of the interplay between the Hurst indices of the underlying fractional Brownian motions and the geometry of the maximum variance curve.

Experimental results

Research questions

• RQ1 What is the exact asymptotic behavior of the supremum of a Gaussian random field when the variance achieves its maximum over a smooth curve rather than a single point?
• RQ2 How do the generalized Pickands and Piterbarg constants emerge in the tail asymptotics of such fields?
• RQ3 What is the role of the Hurst indices and the local correlation structure in determining the decay rate of the tail probability?
• RQ4 How does the non-uniqueness of the maximum variance set affect the asymptotic scaling compared to the unique maximum case?
• RQ5 Can the derived asymptotics be applied to real stochastic processes such as the aggregation of two independent fractional Brownian motions?

Key findings

• The tail probability of the supremum of the Gaussian field satisfies the asymptotic relation $ \mathbb{P}\left\{\sup_{(s,t)\in E}X(s,t)>u\right\} \sim C \cdot u^{2/\alpha_1} \Psi(u) $ as $ u \to \infty $, where $ \Psi(u) $ is the standard normal tail function.
• The constant $ C $ is explicitly expressed in terms of generalized Pickands and Piterbarg constants involving the Hurst indices $ \alpha_1, \alpha_2 \in (0,2] $, with $ \alpha_1 $ governing the decay rate.
• For the case $ \alpha_1 = \alpha_2 = \alpha $, the asymptotic constant is $ \frac{2^{3-2/\alpha}}{\alpha} (\mathcal{H}_\alpha)^2 \int_0^1 (1 - t^\alpha)^{1/\alpha - 1} dt $, which depends on the square of the Pickands constant $ \mathcal{H}_\alpha $.
• When $ \alpha_1 = \alpha_2 = 1 $, the asymptotic constant simplifies to $ 2 \widehat{\mathcal{H}}_1^{1,-1} $, involving a modified Piterbarg constant.
• For $ \alpha_1 < 1 $, the asymptotic behavior is dominated by the term $ u^{4/\alpha_1 - 2} \Psi(u) $, reflecting a slower decay than in the $ \alpha_1 > 1 $ case.
• The results are applied to the supremum of $ X(s+t) - X(s) $, showing that the tail asymptotics for Shepp-type statistics can be derived from the general framework presented.