Truncated Moments for Heavy-Tailed and Related Distribution Classes

Suppose that ξ+ is the positive part of a random variable defined on probability space (Ω,F,P) with distribution function Fξ. When moment Eξ+p order p>0 finite, then truncated F¯ξ,p(x)=min1,Eξp1I{ξ>x}, for all x⩾0, survival or, in other words, tail Fξ,p. In this paper, we examine which regularity properties transfer from Fξ to Fξ,p and The construction describes transformation initial Our results show subclasses heavy-tailed distributions, such as regularly varying, dominatedly consistently varying long-tailed classes, are closed under transformation. We also exponential-like-tailed generalized contain both heavy- light-tailed On hand, demonstrate classes admit inverse closures, i.e., condition belongs one these it follows corresponding class. general, obtained complement known closure classes.