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. Questo
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.

Questo articolo esplora i giri e le loro implicazioni.

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