Truncated Moments for Heavy-Tailed and Related Distribution Classes

• Main Authors: Leipus, R. (Author), Paukštys, S. (Author), Šiaulys, J. (Author)
• Format: Article
• Language: English
• Published: MDPI 2023
• Subjects: alternative moment formula; consistent variation; dominated variation; exponential-like-tailed distribution; heavy-tailed distribution; long-tailed distribution; regular variation; truncated moment
• Online Access: View Fulltext in Publisher; View in Scopus
• ISBN: 22277390 (ISSN)
• DOI: 10.3390/math11092172

Suppose that (Formula presented.) is the positive part of a random variable defined on the probability space (Formula presented.) with the distribution function (Formula presented.). When the moment (Formula presented.) of order (Formula presented.) is finite, then the truncated moment (Formula presented.), defined for all (Formula presented.), is the survival function or, in other words, the distribution tail of the distribution function (Formula presented.). In this paper, we examine which regularity properties transfer from the distribution function (Formula presented.) to the distribution function (Formula presented.) and which properties transfer from the function (Formula presented.) to the function (Formula presented.). The construction of the distribution function (Formula presented.) describes the truncated moment transformation of the initial distribution function (Formula presented.). Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that (Formula presented.) belongs to one of these classes, it follows that (Formula presented.) also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes. © 2023 by the authors.