Asymptotic independence for unimodal densities

TL;DR

This paper provides a simple geometric condition based on the asymptotic shape of level sets of light-tailed densities to determine asymptotic independence of components, extending classical results for Gaussian densities.

Contribution

It introduces a new geometric criterion for asymptotic independence applicable to light-tailed densities, broadening the scope of Sibuya's classical Gaussian result.

Findings

Establishes a sufficient condition for asymptotic independence based on level set shapes.

Extends Sibuya's result to a wider class of light-tailed densities.

Provides a practical approach when distribution functions are unavailable.

Abstract

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (dfs). Dfs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, one can obtain a good geometric image of the asymptotic shape of the level sets of the density. This paper establishes a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.