The distribution of Yule's 'nonsense correlation'
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In 2017, the authors of ernst2017yule explicitly computed the second moment of Yule's "nonsense correlation," offering the first mathematical explanation of Yule's 1926 empirical finding of nonsense correlation. yule1926. The present work closes the final longstanding open question on the distribution of Yule's nonsense correlation ρ:= ∫_0^1 W_1(t)W_2(t) dt - ∫_0^1 W_1(t) dt ∫_0^1 W_2(t) dt/√(∫_0^1 W^2_1(t) dt - (∫_0^1W_1(t) dt)^2)√(∫_0^1 W^2_2(t) dt - (∫_0^1W_2(t) dt)^2) by explicitly calculating all moments of ρ (up to order 16) for two independent Wiener processes, W_1, W_2. These lead to an approximation to the density of Yule's nonsense correlation, apparently for the first time. We proceed 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.
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