For any given multivariate distribution, explicit formulae for the asymptotic covariances of cumulant vectors of the third and the fourth order are provided here. General expressions for cumulants of elliptically symmetric multivariate distributions are also provided. Utilizing these formulae one can extend several results currently available in the literature, as well as obtain practically useful expressions in terms of population cumulants, and computational formulae in terms of commutator matrices. Results are provided for both symmetric and asymmetric distributions, when the required moments exist. New measures of skewness and kurtosis based on distinct elements are discussed, and other applications to independent component analysis and testing are considered.

中文翻译：

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基于累积向量的统计渐近理论及其应用

对于任何给定的多元分布，此处提供了三阶和四阶累积向量的渐近协方差的明确公式。还提供了椭圆对称多元分布的累积量的一般表达式。利用这些公式，可以扩展文献中当前可用的几个结果，并获得关于人口累积量的实用表达式，以及根据交换矩阵的计算公式。当所需的矩存在时，会提供对称和非对称分布的结果。讨论了基于不同元素的偏度和峰度的新度量，并考虑了独立组件分析和测试的其他应用。