[Paper Review] Extremal Behavior of Gaussian Chaos

Enkelejd Hashorva, Dmitry Korshunov+1 more|arXiv (Cornell University)|Jul 22, 2013

Stochastic processes and statistical mechanics5 citations

TL;DR

This paper derives asymptotic expansions for the tail distribution and density of Gaussian chaos $ h(\xi) $, where $ \xi $ is a centered $ d $-dimensional Gaussian vector and $ h $ is a smooth homogeneous function. Using a direct probabilistic asymptotic method, it establishes precise large deviations behavior for extreme values, with applications to determinant of random matrices, Gaussian orthogonal ensemble, and random Gaussian clouds.

ABSTRACT

For a centered $d$-dimensional Gaussian random vector $\xi =(\xi_1,\ldots,\xi_d)$ and a homogeneous function $h:R^d o R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(\xi)$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(\xi)$ and its density at infinity and then discuss possible extensions for some general $\xi$ with polar representation.

Motivation & Objective

• To understand the extreme value behavior of Gaussian chaos $ h(\xi) $, where $ \xi $ is a centered Gaussian vector and $ h $ is a smooth homogeneous function.
• To derive precise asymptotic expansions for the tail distribution and density of $ h(\xi) $ at infinity.
• To analyze challenging instances of Gaussian chaos, including the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble, and the diameter of random Gaussian clouds.
• To extend results to general $ \xi $ with polar representation, broadening applicability beyond i.i.d. Gaussians.

Proposed method

• Applies a direct probabilistic asymptotic method to analyze the tail behavior of $ h(\xi) $, focusing on large deviations in the extreme regime.
• Uses the homogeneity and smoothness of $ h $ to derive asymptotic expansions for the tail probability $ \mathbb{P}(h(\xi) > x) $ as $ x \to \infty $.
• Analyzes the density of $ h(\xi) $ at infinity by studying the decay rate of the density function in the large-deviation regime.
• Employs polar representation of $ \xi $ to generalize results beyond i.i.d. Gaussian components to more general spherically symmetric or elliptically contoured distributions.
• Relies on Laplace's method and saddle-point approximations in the asymptotic analysis of the chaos distribution.
• Establishes connections between the geometry of the level sets of $ h $ and the tail decay rate, using directional derivatives and curvature information.

Experimental results

Research questions

• RQ1 How does the tail distribution of a smooth homogeneous Gaussian chaos $ h(\xi) $ behave as the deviation $ x \to \infty $?
• RQ2 What is the precise asymptotic form of the density of $ h(\xi) $ in the extreme upper tail?
• RQ3 How do the asymptotic behaviors of the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble, and the diameter of a random Gaussian cloud compare?
• RQ4 Can the asymptotic analysis be extended to general $ \xi $ with a polar representation beyond i.i.d. Gaussians?
• RQ5 What role do the geometry and curvature of the level sets of $ h $ play in determining the tail decay rate?

Key findings

• The tail probability $ \mathbb{P}(h(\xi) > x) $ decays asymptotically as $ \exp\left( -\frac{1}{2} \|x\|^{2/k} \cdot \kappa \right) $ up to polynomial corrections, where $ k $ is the degree of homogeneity and $ \kappa $ depends on the geometry of $ h $.
• The density of $ h(\xi) $ at infinity decays exponentially with a rate determined by the maximum of the norm of the gradient of $ h $ on the unit sphere.
• For the determinant of a $ d \times d $ Gaussian matrix, the tail decays as $ \exp\left( -c_d x^{2/d} \right) $ for some constant $ c_d $, matching the general chaos asymptotics.
• The Gaussian orthogonal ensemble exhibits a tail decay rate consistent with the general framework, with explicit dependence on the dimension $ d $.
• The diameter of a random Gaussian cloud in $ \mathbb{R}^d $ shows a tail behavior that aligns with the derived asymptotic expansions, confirming the method's broad applicability.
• The extension to general $ \xi $ with polar representation preserves the asymptotic structure, provided the radial and spherical components are suitably regular.