Xuming He (Chair)

Abstract

In 1926, G. Udny Yule considered the following problem: given a sequence of pairs of random variables $\{X_k,Y_k \}$ ($k=1,2, \ldots, n$), and letting $X_i = S_i$ and $Y_ i= S'_i$ where $S_i$ and $S'_i$ are the partial sums of two independent random walks, what is the distribution of the empirical correlation coefficient

\[\rho_n = \frac{\sum_{i=1}^n S_i S^\prime_i - \frac{1}{n}(\sum_{i=1}^n S_i)(\sum_{i=1}^n S^\prime_i)}{\sqrt{\sum_{i=1}^n S^2_i - \frac{1}{n}(\sum_{i=1}^n S_i)^2}\sqrt{\sum_{i=1}^n (S^\prime_i)^2 - \frac{1}{n}(\sum_{i=1}^n S^\prime_i)^2}}?\]

Yule empirically observed the distribution of this statistic to be heavily dispersed and frequently large in absolute value, leading him to call it ``nonsense correlation.'' This unexpected finding led to his formulation of two concrete questions, each of which would remain open for more than ninety years: (i) Find (analytically) the variance of $\rho_n$ as $n \rightarrow \infty$ and (ii): Find (analytically) the higher order moments and the density of $\rho_n$ as $n \rightarrow \infty$. Ernst, Shepp, and Wyner (\textit{The Annals of Statistics}, 2017) considered the empirical correlation coefficient

\[\rho:= \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - {\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - {\int_0^1W_2(t) dt}^2}}\]

of two \textit{independent} Wiener processes $W_1,W_2$, the limit to which $\rho_n$ converges weakly, as was first shown by P.C.B. Phillips. Using tools from integral equation theory, Ernst et al. (2017) closed question (i) by explicitly calculating the second moment of $\rho$ to be .240522. This talk begins where Ernst et al. (2017) leaves off. I shall explain how we finally succeeded in closing question (ii) by explicitly calculating all moments  of $\rho$ (up to order 16).

This leads, for the first time, to an approximation to the density of Yule's nonsense correlation. I shall then proceed to explain how we were able to explicitly compute higher moments of  $\rho$ when the two independent Wiener processes are replaced by two correlated Wiener processes, two independent Ornstein-Uhlenbeck processes, and two independent Brownian bridges. I will conclude by stating a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes. This result shows that Yule's ``nonsense correlation'' is indeed not ``nonsense'' for stochastic processes which admit stationary distributions.   (Joint work with L.C.G. Rogers at Cambridge and Quan Zhou at Texas A&M).