[Paper Review] Asymptotics of empirical distribution function for Gaussian subordinated arrays with an application to multiple testing

Sylvain Delattre, Étienne Roquain|arXiv (Cornell University)|Oct 9, 2012

Point processes and geometric inequalities1 citations

TL;DR

This paper develops an asymptotic theory for the empirical distribution function (e.d.f.) of non-stationary, high-dimensional Gaussian vector components with dimension-dependent correlation matrices. It shows that the e.d.f.'s behavior depends only on the sequence $\gamma_m = m^{-2} \sum_{i \neq j} \Gamma_{i,j}^{(m)}$, enabling applications to multiple testing in non-stationary, long-range dependent settings.

ABSTRACT

This paper introduces a new framework to study the asymptotical behavior of the empirical distribution function (e.d.f.) of Gaussian vector components, whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. Hence, by contrast with the existing literature, the vector is not assumed to be stationary. Rather, we make a vanishing second order assumption ensuring that the covariance matrix $\Gamma^{(m)}$ is not too far from the identity matrix, while the behavior of the e.d.f. is affected by $\Gamma^{(m)}$ only through the sequence $\gamma_m=m^{-2} \sum_{i eq j} \Gamma_{i,j}^{(m)}$, as $m$ grows to infinity. This result recovers some of the previous results for stationary long-range dependencies while it also applies to various, high-dimensional, non-stationary frameworks, for which the most correlated variables are not necessarily next to each other. Finally, we present an application of this work to the multiple testing problem, which was the initial statistical motivation for developing such a methodology.

Motivation & Objective

• To extend the asymptotic theory of empirical distribution functions beyond stationary Gaussian processes.
• To model high-dimensional, non-stationary Gaussian arrays where correlations are not localized or stationary.
• To identify a sufficient statistic $\gamma_m = m^{-2} \sum_{i \neq j} \Gamma_{i,j}^{(m)}$ that governs the e.d.f. asymptotics despite dimension-dependent correlation matrices.
• To provide a theoretical foundation for multiple testing procedures in complex, non-stationary high-dimensional settings.

Proposed method

• Introduce a vanishing second-order condition on the correlation matrix $\Gamma^{(m)}$, ensuring it remains close to the identity matrix as $m \to \infty$.
• Define the key sequence $\gamma_m = m^{-2} \sum_{i \neq j} \Gamma_{i,j}^{(m)}$ to capture the aggregate dependence structure.
• Establish the asymptotic distribution of the empirical distribution function using only $\gamma_m$, independent of the full correlation matrix structure.
• Apply the asymptotic result to multiple testing by deriving a test statistic whose null distribution depends on $\gamma_m$.
• Use concentration and weak convergence techniques to prove convergence of the e.d.f. under the $\gamma_m$-based normalization.
• Demonstrate that the method recovers known results for stationary long-range dependent processes as special cases.

Experimental results

Research questions

• RQ1 How does the empirical distribution function behave asymptotically for high-dimensional, non-stationary Gaussian arrays with dimension-dependent correlation matrices?
• RQ2 Can the asymptotic behavior of the e.d.f. be characterized using a single scalar sequence $\gamma_m$ rather than the full correlation matrix?
• RQ3 Does the proposed framework include stationary long-range dependent processes as special cases?
• RQ4 Can this theory be applied to multiple testing in high-dimensional, non-stationary settings?

Key findings

• The asymptotic distribution of the empirical distribution function depends solely on the sequence $\gamma_m = m^{-2} \sum_{i \neq j} \Gamma_{i,j}^{(m)}$, not on the full structure of $\Gamma^{(m)}$.
• The framework generalizes existing results for stationary long-range dependent processes by allowing non-stationarity and non-localized correlations.
• The method applies to high-dimensional settings where the most correlated variables are not necessarily adjacent or clustered.
• The asymptotic theory supports valid inference in multiple testing problems under weak dependence assumptions captured by $\gamma_m$.