[Paper Review] Extrema of locally stationary Gaussian fields on growing manifolds

Wanli Qiao, Wolfgang Polonik|arXiv (Cornell University)|Oct 23, 2015

Hydrology and Drought Analysis27 references3 citations

TL;DR

This paper establishes the asymptotic distribution of extreme values for locally stationary Gaussian fields on rescaled, growing manifolds as the scaling parameter $ h \to 0 $. By generalizing classical results of Bickel and Rosenblatt (1973a) and Mikhaleva and Piterbarg (1997), it derives a limit theorem for the supremum of such fields, showing convergence to a Weibull-type extreme value distribution with a geometric intensity determined by the manifold's dimension and local covariance structure.

ABSTRACT

We consider a class of non-homogeneous, continuous, centered Gaussian random fields $\{X_h(t), t \in {\cal M}_h;\,0 < h \le 1\}$ where ${\cal M}_h$ denotes a rescaled smooth manifold, i.e. ${\cal M}_h = \frac{1}{h} {\cal M},$ and study the limit behavior of the extreme values of these Gaussian random fields when $h$ tends to zero, which means that the manifold is growing. Our main result can be thought of as a generalization of a classical result of Bickel and Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).

Motivation & Objective

• To derive the limiting distribution of the supremum of non-homogeneous, locally stationary Gaussian fields on rescaled manifolds as the manifold grows.
• To generalize classical extreme value results for Gaussian processes on intervals or cubes to the case of smooth, compact, $ r $-dimensional manifolds.
• To provide a theoretical foundation for uniform inference on geometric objects such as ridges, level sets, and density boundaries in nonparametric statistics.
• To establish a uniform asymptotic result for excursion probabilities over manifolds that are scaled by $ h \to 0 $, relevant for confidence band construction.
• To extend the applicability of extreme value theory to non-stationary, manifold-indexed Gaussian fields with local isotropy properties.

Proposed method

• Formalize a class of centered, continuous, locally stationary Gaussian fields $ \{X_h(t), t \in \mathcal{M}_h\} $, where $ \mathcal{M}_h = \frac{1}{h}\mathcal{M} $ is a rescaled smooth manifold.
• Use a local expansion of the covariance structure around each point on the manifold to model local stationarity via a matrix function $ D_t $.
• Apply a triangulation and meshing technique to discretize the rescaled manifold $ \mathcal{M}_h $, approximating the supremum over continuous fields by maxima over discrete grids.
• Leverage the Pickands-type extremal index and the Pickands–Berman method for Gaussian fields with $ \alpha $-regular covariance decay $ r(t) = 1 - \|t\|^\alpha + o(\|t\|^\alpha) $.
• Use the limit result from Piterbarg (1996) and Mikhaleva–Piterbarg (1997) on excursion probabilities over fixed manifolds, extended to the growing case via scaling.
• Establish uniform convergence over families of Jordan measurable sets with bounded Hausdorff measure, ensuring robustness across manifold shapes and sizes.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the supremum of a locally stationary Gaussian field on a rescaled manifold as the scaling parameter $ h \to 0 $?
• RQ2 How does the extreme value behavior of such fields depend on the local geometry and covariance structure of the underlying manifold?
• RQ3 Can the classical Bickel–Rosenblatt result on suprema of kernel density estimators be generalized to fields indexed on manifolds?
• RQ4 What is the asymptotic behavior of the excursion probability $ \mathbb{P}(\sup_{t \in \mathcal{M}_h} X_h(t) > x) $ as $ h \to 0 $, and how does it scale with the manifold's dimension and curvature?
• RQ5 Under what conditions does the limit of the normalized supremum distribution converge to a non-degenerate Weibull-type distribution?

Key findings

• The limit of the normalized supremum distribution is $ \mathbb{P}(\sup_{t \in \mathcal{M}} X(t) > x) \sim x^{2r/\alpha} \Psi(x) \cdot H_\alpha^{(r)} \int_{\mathcal{M}} \|D_s M_s\|_r \, ds $ as $ x \to \infty $, where $ r $ is the dimension of the manifold.
• The asymptotic scaling factor $ x^{2r/\alpha} \Psi(x) $ captures the interplay between the local Hölder regularity $ \alpha $ of the covariance and the intrinsic dimension $ r $ of the manifold.
• The limit intensity is determined by the integral of the local scaling matrix $ D_s $ over the manifold, weighted by the tangent space orientation via $ M_s $, the orthonormal basis matrix of the tangent space at $ s $.
• The convergence is uniform over families of Jordan measurable $ r $-dimensional sets with bounded $ r $-dimensional Hausdorff measure, ensuring robustness for statistical inference.
• The result generalizes Bickel and Rosenblatt (1973a) from intervals to manifolds and extends Mikhaleva and Piterbarg (1997) from fixed to growing manifolds.
• The extreme value limit is non-degenerate and of Weibull type, with the normalization constants $ a_h $ and $ b_h $ in (1.1) implicitly determined by the $ x^{2r/\alpha} \Psi(x) $ scaling.