On the non-closure under convolution for strong subexponential distributions

Abstract

In this paper, we consider the convolution closure problem for the class of strong subexponential distributions, denoted as S*. First, we show that, if F, G ∈ L, then inclusions of F*G, FG, and pF + (1 – p)G for all (some) p ∈ (0; 1) into the class S* are equivalent. Then, using examples constructed by Klüppelberg and Villasenor [The full solution of the convolution closure problem for convolution-equivalent distributions, J. Math. Anal. Appl., 41:79–92, 1991], we show that S* is not closed under convolution.

Randomly Stopped Sums, Minima and Maxima for Heavy-tailed and Light-Tailed Distributions

“…Similarly, we are interested when F X (ν) , F X (ν) , F S (ν) and F S (ν) are heavy-tailed or light tailed. For various distribution classes, similar questions were studied in [1][2][3][4][5][6][7][8][9][10], [11][12][13][14][15][16][17][18][19][20], [21][22][23][24][25][26][27][28][29][30]. We mention also the paper [31], where two independent heavy-tailed r.v.s, such that their minimum is not heavy tailed, were constructed.…”

“…of the proposition is proved. According to the inequality(17) and Lemma 2,F X (ν) ∈ H c if F X (κ) ∈ H c . Since κ is finite, conditions F X k ∈ H c , k ∈ {1, 2, .. .…”