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Asymptotics of Yule's nonsense correlation for Ornstein-Uhlenbeck paths: The correlated case

Soukaina Douissi, Philip Ernst, Frederi Viens

TL;DR

This work analyzes the continuous-time Yule nonsense correlation for correlated Ornstein–Uhlenbeck paths, deriving sharp Wiener-space–based CLTs for the empirical correlation $\rho(T)$ as the horizon $T$ grows. The authors decompose the numerator into second-chaos components, obtain explicit cumulant control via the optimal fourth moment theorem, and couple these with LLNs for the denominator to obtain a full Gaussian limit for $\rho(T)$ with a known rate. They translate these asymptotics into two statistically powerful independence tests—one based on the empirical correlation and another on its numerator—providing explicit rejection regions and finite-sample error bounds; they also illustrate an infinite-dimensional application to stochastic PDEs, showing how independence tests can be boosted by using multiple Fourier modes. The results yield practical, quantitative tools for testing independence in continuous-time Gaussian models and open avenues for extending Wiener-chaos methods to broader stationary Gaussian and infinite-dimensional settings.

Abstract

We study the continuous-time version of the empirical correlation coefficient between the paths of two possibly correlated Ornstein-Uhlenbeck processes, known as Yule's nonsense correlation for these paths. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. An application to independence of two observations of solutions to the stochastic heat equation is given, with excellent asymptotic power properties using merely a small number of the solutions' Fourier modes.

Asymptotics of Yule's nonsense correlation for Ornstein-Uhlenbeck paths: The correlated case

TL;DR

This work analyzes the continuous-time Yule nonsense correlation for correlated Ornstein–Uhlenbeck paths, deriving sharp Wiener-space–based CLTs for the empirical correlation as the horizon grows. The authors decompose the numerator into second-chaos components, obtain explicit cumulant control via the optimal fourth moment theorem, and couple these with LLNs for the denominator to obtain a full Gaussian limit for with a known rate. They translate these asymptotics into two statistically powerful independence tests—one based on the empirical correlation and another on its numerator—providing explicit rejection regions and finite-sample error bounds; they also illustrate an infinite-dimensional application to stochastic PDEs, showing how independence tests can be boosted by using multiple Fourier modes. The results yield practical, quantitative tools for testing independence in continuous-time Gaussian models and open avenues for extending Wiener-chaos methods to broader stationary Gaussian and infinite-dimensional settings.

Abstract

We study the continuous-time version of the empirical correlation coefficient between the paths of two possibly correlated Ornstein-Uhlenbeck processes, known as Yule's nonsense correlation for these paths. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. An application to independence of two observations of solutions to the stochastic heat equation is given, with excellent asymptotic power properties using merely a small number of the solutions' Fourier modes.

Paper Structure

This paper contains 13 sections, 21 theorems, 184 equations.

Table of Contents

  1. Introduction and Setup
  2. Elements of the analysis on Wiener space
  3. Fluctuations of $\rho (T)$ under $H_{a}$ : CLTs and rates of convergence
  4. Gaussian fluctuations of the numerator $Y_{12}(T)$
  5. Law of large numbers of $\rho(T)$ under $H_a$ :
  6. Gaussian fluctuations of $\rho(T)$ under $H_a$:
  7. Some statistical applications of the Gaussian asymptotics of $\rho (T)$ under $H_{a}$
  8. Testing independence of $X_{1}$ and $X_{2}$: Rejection region and power of the test
  9. A test of independence using the numerator :
  10. Future directions and and application
  11. Future directions
  12. An application: testing independence with stochastic PDEs
  13. Appendix

Key Result

Theorem 1

There exists a constant $C(\theta,r)$ depending on $\theta$ and $r$ such that where $\sigma_{r,\theta} := \left(\frac{1}{2 \theta^3} \left(\frac{1}{2}+\frac{r^2}{2}\right)\right)^{1/2}.$ In particular,

Theorems & Definitions (24)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • Theorem 8
  • Remark 9
  • Proposition 10
  • ...and 14 more