Abstract
Let $\mathbf{x} = (x_1,\cdots, x_n)'$ be a random vector in $R^n$. Two characterizations of normality are given. One involves the existence of two linear combinations of the $\{x_j\}$ that are independent in every coordinate system. The other, which is actually a consequence of the first, assumes that $\mathbf{x}$ obeys a linear model with spherical errors and involves sufficiency of the least-squares estimator.
Citation
Robert H. Berk. "Sphericity and the Normal Law." Ann. Probab. 14 (2) 696 - 701, April, 1986. https://doi.org/10.1214/aop/1176992538
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