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Tests of independence for pairs of paths of non-stationary Gaussian processes

Philip A. Ernst, Frederi G. Viens, Shuo Yan

TL;DR

The paper develops a rigorous asymptotic theory for testing independence between two jointly Gaussian non-stationary paths, focusing on Brownian motion and fractional Brownian motion with $H>1/2$, through the discretized empirical Pearson correlation $\rho_n$. It shows that $n(\rho_n-\rho)$ admits a decomposition into a path-functional term $\mu$ and a Gaussian fluctuation $\mathcal{Z}$ (conditionally on the paths), with explicit limiting variances $\sigma^r$ for Brownian motion and $\sigma_H$ for fBm; a delta-method yields a central limit theorem for the empirical correlation $\rho_n$ as well. The authors also provide discrete analogs, practical testing procedures, and Monte Carlo-based estimates of the variance terms to enable implementation with high-frequency data. Numerics corroborate the theoretical limits for both continuous and discrete sampling, highlighting the method's robustness to non-stationarity and long-range dependence. Overall, the work offers a principled path-based independence test for non-stationary Gaussian processes with concrete, implementable limiting distributions and variance formulas.

Abstract

In the current work, we provide theoretical results for testing (in)dependence between pairs of paths of most commonly studied non-stationary Gaussian processes - standard Brownian motion and fractional Brownian motion (fBm). Please see the PDF version of the paper for a full abstract.

Tests of independence for pairs of paths of non-stationary Gaussian processes

TL;DR

The paper develops a rigorous asymptotic theory for testing independence between two jointly Gaussian non-stationary paths, focusing on Brownian motion and fractional Brownian motion with , through the discretized empirical Pearson correlation . It shows that admits a decomposition into a path-functional term and a Gaussian fluctuation (conditionally on the paths), with explicit limiting variances for Brownian motion and for fBm; a delta-method yields a central limit theorem for the empirical correlation as well. The authors also provide discrete analogs, practical testing procedures, and Monte Carlo-based estimates of the variance terms to enable implementation with high-frequency data. Numerics corroborate the theoretical limits for both continuous and discrete sampling, highlighting the method's robustness to non-stationarity and long-range dependence. Overall, the work offers a principled path-based independence test for non-stationary Gaussian processes with concrete, implementable limiting distributions and variance formulas.

Abstract

In the current work, we provide theoretical results for testing (in)dependence between pairs of paths of most commonly studied non-stationary Gaussian processes - standard Brownian motion and fractional Brownian motion (fBm). Please see the PDF version of the paper for a full abstract.

Paper Structure

This paper contains 22 sections, 24 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Discrete version of the results
  4. Hypothesis testing
  5. Results on standard Brownian motions
  6. Representation of the empirical variance as iterated Itô integrals
  7. Weak convergence of the integrands
  8. Weak convergence of the covariance and correlation: Proof of Theorem \ref{['sthm:cov']}
  9. Covariance: independence case
  10. Covariance: general case
  11. Empirical correlation
  12. Results on fractional Brownian motions
  13. Representation of the covariance as iterated Stratonovich integrals
  14. Convergence of the integrands
  15. L2 convergence to zero
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mathcal{W}_1$ and $\mathcal{W}_2$ be two standard Brownian motions that are jointly Gaussian with constant correlation coefficient $r$. Let $\mathcal{X}^{i,j}$ and $\mathcal{X}^{i,j}_n$ for $i,j\in\{1,2\}$ and corresponding $\rho$ and $\rho_n$ be defined as in def:empirical with $X_i=\mathcal{

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 33 more