On closure and factorization properties of subexponential and related distributions

Abstract

Abstract For a distribution function F on [0, ∞] we say F ∈ if {1 – F (2) ( x )}/{1 – F ( x )}→2 as x →∞, and F ∈ , if for some fixed γ > 0, and for each real , lim x→∞ {1 – F ( x + y )}/{1 – F ( x )} ═ e – n . Sufficient conditions are given for the statement F ∈ F * G ∈ and when both F and G are in y it is proved that F * G ∈ pF + 1(1 – p ) G ∈ for some (all) p ∈(0,1). The related classes ℒ t are proved closed under convolutions, which implies the closure of the class of positive random variables with regularly varying tails under multiplication (of random variables). An example is given that shows to be a proper subclass of ℒ 0 .