Assume that $\{X_{k}\}_{k=1}^{\infty}$ and $\{Y_{k}\}_{k=1}^{\infty}$ are two independent sequences of independent standard Gaussian random variables. We consider $$ \theta_{n} := \frac{ \frac{1}{n} \sum_{i=1}^{n} S_i T_i - \frac{1}{n^2} (\sum_{i=1}^{n} S_i) (\sum_{i=1}^{n} T_i) }{ \sqrt{ \frac{1}{n} \sum_{i=1}^{n} S_i^2 - \frac{1}{n^2} (\sum_{i=1}^{n} S_i)^2 } \sqrt{ \frac{1}{n} \sum_{i=1}^{n} T_i^2 - \frac{1}{n^2} (\sum_{i=1}^{n} T_i)^2 }}, $$ which is Yule's "nonsense correlation" in the discrete version. We obtain its moments by calculating the corresponding joint moment generating function explicitly. In particular, the variances of $\theta_{n}$ are quite large. The limit of the variances is nearly $0.5$ even though as $n$ tends to $\infty$. This fact reproves its distribution is heavily dispersed and is frequently large in absolute value. So it is problematic to use $\theta_{n}$ as a statistic to test the independence between two random walks.
翻译：假设美元=X ⁇ k ⁇ k ⁇ k=1 ⁇ ginfty}美元和美元=1 ⁇ k ⁇ k=1 ⁇ infty}美元是独立标准高斯随机变量的两个独立序列。 我们认为 $=\\\ t ⁇ n} =\ frac{\ {1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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