On closure and factorization properties of subexponential and related distributions

Abstract

For a distribution function F on [0, ℮) we say FεL if {1 — F(2)(x)}/{1 —f(x)}→2 as x→∞, and FεLy if for some fixed y > 0, and for each real y, limx→∞ {1 — F(x+y)}/{1 — F(x)} = e-yy. Sufficient conditions are given for the statement a FεL⇔F*GεL, nd when both F and G are in L it is proved that F*GεL⇔pF+(l-p)GεL for some (all)pε(0,1). The related classes Ly 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 L to be a proper subclass of L o. © 1980, Australian Mathematical Society. All rights reserved.