Asymptotic Expansions for Heavy-Tailed Data

Heavy-tailed distributions are present in the characterization of different modern systems such as high-resolution imaging, cloud computing, and cognitive radio networks. Commonly, the cumulants of these distributions cannot be defined from a certain order, and this restricts the applicability of traditional methods. To fill this gap, the present letter extends the traditional Edgeworth and Cornish-Fisher expansions, which are based on the cumulants, to analogous asymptotic expansions based on the log-cumulants. The proposed expansions inherit the capability of log-cumulants to characterize heavy-tailed distributions and parallel traditional expansions. Thus, they are readily implemented. Interestingly, the proposed expansions are applicable for light-tailed distributions as well.