On closure and factorization properties of subexponential and related distributions

Abstract

For a distribution function F on [0, oo) we say FeSf if {1 -

The asymptotic hazard rate of sums of discrete random variables

“…Proposition 1 is related to a classical convolution property of subexponential and related distributions; in particular Theorem 3 of Embrechts and Goldie (1980); more recent formulations can be found in Block et al (2014Block et al ( , 2015. In this article, we focus on integer-valued distributions.…”

“…In addition, the inequality on the last line of page 253 of Embrechts and Goldie (1980) is not correct. In what follows we correct (and even simplify) that proof and have it apply to our discrete setting.…”

Asymptotics in the symmetrization inequality

“…Pakes [11] or Teugels [12]), we may assume that X, Y are a.s. non-negative. In this case the lemma is theorem 2 in Embrechts and Goldie [4].…”

“…In Embrechts and Goldie [4], Thm. 1, it is proved that if F ∈ L , G ∈ S , and sup x G(x)/F (x) < ∞, then F ∈ S ⇔ F * G ∈ S .…”

A Note on the Closure of Convolution Power Mixtures (Random Sums) of Exponential Distributions

“…As with the proof of Cline [4, Lemma 2.3(ii)], and half the proof of Cline [5], our arguments take off from the fundamental estimate (5) below from the proof of Embrechts and Goldie [6] that L(α) is closed under convolution.…”