Philip A. Ernst - Yule's ``nonsense correlation'' solved: Part II
| Dates: | 7 February 2024 |
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| Times: | 16:00 - 17:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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Philip A. Ernst (Imperial College London) will speak at the Probability seminar.
In 1926, G. Udny Yule considered the following problem: given a sequence of pairs of random variables {X_k, Y_k} for k from 1 to n, and letting X_i = S_i and Yi = S'_i where S_i and S'_i are the partial sums of two independent random walks, what is the distribution of their empirical correlation coefficient p_n?
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 p_n when n is large and (ii) find (analytically) the higher order moments and the density of p_n when n is large.
Ernst, Shepp, and Wyner (The Annals of Statistics, 2017) considered the empirical correlation coefficient of two independent Wiener processes W_1, W_2, the limit to which p_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 p to be 0.240522. This talk begins where Ernst et al. (2017) leave off. I shall explain how we finally succeeded in closing question (ii) by explicitly calculating all moments of p (up to order 16). This leads, for the first time, to an approximation of the density of Yule's nonsense correlation. I shall then proceed to explain how we were able to explicitly compute higher moments of p when the two independent Wiener processes are replaced by two correlated Wiener processes, two independent Ornstein-Uhlenbeck processes, and two independent Brownian bridges. This talk concludes 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'' when considering stochastic processes admitting stationary distributions. This is joint work with L.C.G. Rogers (Cambridge) and Quan Zhou (Texas A&M). The paper can be found at https://arxiv.org/pdf/1909.02546.pdf (under minor revision, Bernoulli).
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