Truncated Moments for Heavy-Tailed and Related Distribution Classes

Suppose that ξ + is the positive part of a random variable defined on the probability space ( Ω , F , P ) with the distribution function F ξ . When the moment E ξ + p of order p > 0 is finite, then the truncated moment F ¯ ξ , p ( x ) = min 1 , E ξ p 1 I { ξ > x } , defined for all x ⩾ 0 , is the survival function or, in other words, the distribution tail of the distribution function F ξ , p . In this paper, we examine which regularity properties transfer from the distribution function F ξ to the distribution function F ξ , p and which properties transfer from the function F ξ , p to the function F ξ . The construction of the distribution function F ξ , p describes the truncated moment transformation of the initial distribution function F ξ . 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 F ξ , p belongs to one of these classes, it follows that F ξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.