Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44, 2089–2097; p. 2091) speculated that ‘the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·’. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor’s Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances.

中文翻译：

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重尾分布、相关性、峰态和泰勒波动标度定律

Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44, 2089–2097; p. 2091) 推测“[随机变量，rvs] 之间的依赖性可能会被其边缘尾部的沉重感压倒···”。我们给出了支持这种推测的统计模型的例子。虽然在自然条件下，规则变化 (RV) rvs 的样本相关性收敛到通常随机的极限，但当 rvs 是大于任意（但不完美）正相关或负相关正态之一的幂的倒数时，该极限为零。令人惊讶的是，这些 RV rvs 乘以样本大小的样本相关性在负半线上具有限制分布。我们证明了 RV rvs 的泰勒定律（幂律方差函数）的渐近缩放是一个常数，对于独立和同分布的观测值与大于任意（但不完全）正相关正态之一的幂的倒数相同，无论这些幂是相同还是不同。相关性和异质性不影响渐近标度。我们类似地分析重尾数据的样本峰度。我们表明，在具有重尾预测器和噪声的线性模型中，斜率的最小二乘估计器出乎意料地比它们具有有限方差时收敛得更快。