Convolution and convolution-root properties of long-tailed distributions

Abstract

We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with "plus" and/or "max" operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavytailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called "generalised subexponential" are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie [10,12] for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Lévy measure is neither long-tailed nor generalised subexponential.

Two Non-closure Properties on the Class of Subexponential Densities

“…They also prove in [4] that if L(γ ) ∩ P + with γ > 0 is closed under convolution roots, then S(γ ) ∩ P + with γ > 0 is closed under convolution roots. However, Shimura and Watanabe [12] prove that the class L(γ ) with γ ≥ 0 is not closed under convolution roots, and we find that Xu et al [16] show the same conclusion in the case γ = 0. Pakes [10] and Watanabe [13] show that S(γ ) with γ > 0 is closed under convolution roots in the class of infinitely divisible distributions on R. It is still open whether the class S(γ ) with γ > 0 is closed under convolution roots.…”

On the almost decrease of a subexponential density

“…For n ∈ N + , we denote any two adjacent numbers in set {a n , b n , a n+1 +bn 2 , a n+1 } by c n and d n . By the method of Lemma 4.1 in Xu et al [29] and (2.3), we know that, for any fixed constant t ∈ R + and variable x ∈ R + , there is a positive integer n such that…”

Randomly stopped maximum and maximum of sums with consistently varying distributions

“…The random closure results for class D can be found in [7,16], and for the class L, in [1,16,19,20]. The random closure results for the class C can be derived from the results of [14].…”

“…The random closure results for the class C can be derived from the results of [14]. We note that in [7,14,20], the case of not necessarily identically…”