Скачать с ютуб Frederi Viens (Rice University): Analysis on Wiener space for Yule’s correlation statistic в хорошем качестве

Frederi Viens (Rice University): Analysis on Wiener space for Yule’s correlation statistic

The empirical correlation statistic for two time series of length $n$ is known as Pearson’s correlation. As $n$ tends to infinity, it is known to converge to the true correlation coefficient for the pair of models from whence the series came, as long as these models satisfy some mild stationarity and weak memory conditions. This certainly works well in practice when the series are actually i.i.d. measurements, and applied scientists the world over are aware of this. Things start to go sideways when the memory among datapoints in each sequence is much stronger (the normality of asymptotic fluctuations get destroyed), and even worse, when the stationarity assumption is significantly violated. Famously, convergence to the true correlation will fail dramatically in the case of random walks, as was noticed in 1926 by G. Udny Yule using empirical calculations (done by hand!). The empirical correlation, then known as “Yule’s nonsense correlation” statistic, is asymptotically diffuse, over the entire interval (−1,1). Many decades later, one still runs into vexing instances of applied scientists who draw incorrect attribution conclusions based on invalid inference about correlations of time series, including, arguably, in climate science, in ignorance of Yule’s original observation. We will describe the mathematical question of understanding the asymptotics of Yule’s correlation for random walks, including an explicit expression for its variance when the random walks are Gaussian, and a surprisingly rapid rate of convergence to a limiting diffuse “nonsense correlation” object. These results, based on Wiener chaos calculus, appeared in a paper with Philip Ernst and Dongzhou Huang, in Stochastic Processes and their Applications, in April 2023. We will mention work in progress and a conjectured framework for an exotic conditional central limit theorem with interesting practical implications. For instance, for modeling questions in paleoclimatology, this framework would allow a more principled methodology for climate change attribution, with better robustness properties with respect to model misspecification. The same framework could also be used to develop a metric for assessing the consistency, broadly construed, of pairs of so-called climate ensembles. Time permitting, we will discuss informally our reporting, in a new paleo-climatology preprint with J. Emile-Geay, G. Hakim, F. Zhu, and D. Amrhein, of some challenging mathematical questions which could explain why these and other robust tools are severely needed in this field of application.