New Frontiers in Statistical Inference for Stochastic Processes
I will first briefly review "Yule's 'nonsense correlation' solved!" (The Annals of Statistics, 2017). Consider the standard empirical correlation $\rho_n$, which is defined for two related series of data of length $n$ using the standard Pearson correlation statistic. This empirical correlation is known as Yule's "nonsense correlation" in honor of the British statistician G. Udny Yule, who in 1926 described the phenomenon by which empirical correlation fails to gauge independence of data series for random walks and for other time series. For the case of two independent and identically distributed random walks independent from each other, Yule empirically observed that the distribution of the empirical correlation is not concentrated around 0; rather, it is "volatile" in the sense that its distribution is heavily dispersed and is frequently large in absolute value. This well-documented effect was ignored by many scientists over the decades, up to the present day, even sparking recent controversies in climate-change attribution. Since the 1960s, some probabilists have wanted to eliminate any possible ambiguity about the issue by computing the variance of the continuous-time version $\rho$ of Yule's nonsense correlation, based on the paths of two independent Wiener processes. The problem would remain open for over ninety years until we finally closed it in our 2017 paper.
I will then turn to speaking about our subsequent success in explicitly calculating all moments of $\rho$ for two independent Brownian motions. Our solution leads to the first approximation to the density of Yule's nonsense correlation. We are also able to explicitly compute higher moments of Yule's nonsense correlation when the two independent Wiener processes are replaced by two correlated Wiener processes, two independent Ornstein-Uhlenbeck processes, and two independent Brownian bridges. We then consider extending the definition of $\rho$ to the time interval $[0,T]$ for any $T>0$ and prove a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes. All of these aforementioned results appear in our preprint entitled "The distribution of Yule's nonsense correlation" (under review, https://arxiv.org/pdf/1909.02546.pdf).
Finally, I will then discuss present work in building asymptotically exact and powerful tests of independence for pairs of independent Ornstein-Uhlenbeck processes and for other stationary Gaussian processes. Many of the methods of proof are drawn from Wiener chaos analysis, a simplified way of implementing the so-called Malliavin calculus for random variables depending in a polynomial way on finite or infinite dimensional Gaussian vectors such as Wiener processes. Time permitting, I also hope to speak about some initial leads in building tests of independence for pairs of nonstationary processes, in particular processes with long memory such as the so-called fractional Brownian motion. I will conclude with some concrete applications of our work to the study of weather and climate extremes.
Back to Day 1
hide forever | hide once