“Yule’s ‘Nonsense Correlation’ Solved!”, Philip Ernst, Larry Shepp, Abraham Wyner2016-08-14 (probability theory)⁠:

In this paper, we resolve a long-standing open statistical problem. The problem is to mathematically confirm Yule’s1926 empirical finding of “nonsense correlation”. We do so by analytically determining the second moment [variance] of the empirical correlation coefficient:

of two independent Wiener processes, W₁ & W₂.

Using tools from Fredholm integral equation theory, we successfully calculate the second moment of θ to obtain a value for the standard deviation of θ of nearly 0.5. The “nonsense” correlation, which we call “volatile” correlation, is volatile in the sense that its distribution is heavily dispersed and is frequently large in absolute value. It is induced because each Wiener process is “self-correlated” in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated.

In addition to providing an explicit formula for the second moment of θ, we offer implicit formulas for higher moments of θ.