In this short note, we extend a classical test of subsphericity, based on the first two moments of the eigenvalues of the sample covariance matrix, to the high-dimensional regime where the signal eigenvalues of the covariance matrix diverge to infinity and either $p/n \rightarrow 0$ or $p/n \rightarrow \infty$. In the latter case we further require that the divergence of the eigenvalues is suitably fast in a specific sense. Our work can be seen to complement that of Schott (2006) who established equivalent results in the regime $p/n \rightarrow \gamma \in (0, \infty)$. Simulations are used to demonstrate the results, providing also evidence that the test might be further extendable to a wider asymptotic setting.

翻译：在这个简短的注释中,我们根据样本共变矩阵的均值的前两个时刻,将子球性古典测试扩大到高维系统,即共变矩阵的信号的均值与无限值相差,要么是$p/n\rightrow 0美元,或者美元/p/n\rightrow =fty$。在后一种情况下,我们进一步要求,在特定意义上,电子值的差异是适当的快速的。我们的工作可以被看成是对Schott(2006年)的补充,Schott(2006年)在系统中建立了等值的结果 $p/n\rightrow\gamma \ in (0,\infty) $。模拟用于展示结果,还提供证据,证明测试可以进一步扩展到更广泛的测试环境。