Abstract
In this paper, we study the distribution of the so-called “Yule’s nonsense correlation statistic” on a time interval for a time horizon , when T is large, for a pair of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to:
where the random variables , are defined as
We assume and have the same drift parameter . We also study the asymptotic law of a discrete-type version of , where above are replaced by their Riemann-sum discretizations. In this case, conditions are provided for how the discretization (in-fill) step relates to the long horizon T. We establish identical normal asymptotics for standardized and its discrete-data version. The asymptotic variance of is . We also establish speeds of convergence in the Kolmogorov distance, which are of Berry-Esséen-type (constant*) except for a factor. Our method is to use the properties of Wiener-chaos variables, since and its discrete version are comprised of ratios involving three such variables in the 2nd Wiener chaos. This methodology accesses the Kolmogorov distance thanks to a relation which stems from the connection between the Malliavin calculus and Stein’s method on Wiener space.
Funding Statement
The first and third authors’ research was partially supported by the US NSF award DMS 1811779.
Acknowledgements
We would like to thank the referee for her/his/their careful reading, constructive remarks and useful suggestions.
Citation
Soukaina Douissi. Khalifa Es-Sebaiy. Frederi Viens. "Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: A Wiener chaos approach." Electron. J. Statist. 16 (1) 3176 - 3211, 2022. https://doi.org/10.1214/22-EJS2021
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