Extreme Eigenvalues of Nonlinear Correlation Matrices with Applications to Additive Models

TL;DR

This paper investigates the behavior of extreme eigenvalues in nonlinear correlation matrices, extending classical results to Gaussian vectors, processes, and additive models, with implications for stochastic dependence measurement.

Contribution

It generalizes the understanding of maximum nonlinear correlation to complex structures like Gaussian vectors, processes, and symmetric functions, with applications to additive regression models.

Findings

Extended eigenvalue results to Gaussian vectors and processes

Connected nonlinear correlation to additive regression models

Provided theoretical insights into stochastic dependence measures

Abstract

The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.