The full solution of the convolution closure problem for convolution- equivalent distributions. (English) Zbl 0731.60013

A distribution function F belongs to the class S(\(\gamma\)) with \(\gamma\geq 0\) if
(i) \(\lim_{x\to \infty}(1-F^{2*}(x))/(1-F(x))=2\hat f(\gamma)<\infty\),
(ii) \(\lim_{x\to \infty}(1-F(x-y))/(1-F(x))=e^{\gamma y}\forall y\in {\mathbb{R}},\)
where \(F^{2*}\) is the convolution square and \(\hat f\) is the moment generating function of F. Such distribution functions are called convolution-equivalent. We construct an example that proves that none of these classes \({\mathcal S}(\gamma)\) is closed under convolution.


60E05 Probability distributions: general theory
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[1] Chover, J; Ney, P; Wainger, S, Functions of probability measures, J. analyse math., 26, 255-302, (1973) · Zbl 0276.60018
[2] Chover, J; Ney, P; Wainger, S, Degeneracy properties of subcritical branching processes, Ann. probab., 1, 663-673, (1973) · Zbl 0387.60097
[3] Cline, D.B.H, Convolution tails, product tails and domains of attraction, Probab. theory related fields, 72, 529-557, (1986) · Zbl 0577.60019
[4] Cline, D.B.H, Convolutions of distributions with exponential and subexponential tails, J. austral. math. soc. ser. A, 43, 347-365, (1987) · Zbl 0633.60021
[5] Embrechts, P; Goldie, C.M, On closure and factorization properties of subexponential and related distributions, J. austral. math. soc. ser. A, 29, 243-256, (1980) · Zbl 0425.60011
[6] Embrechts, P; Goldie, C.M, On convolution tails, Stochastic process. appl., 13, 263-278, (1982) · Zbl 0487.60016
[7] Feller, W, ()
[8] Klüppelberg, C, Subexponential distributions and integrated tails, J. appl. probab., 25, 132-141, (1988) · Zbl 0651.60020
[9] Klüppelberg, C, Subexponential distributions and characterizations of related classes, Probab. theory related fields, 82, 259-269, (1989) · Zbl 0687.60017
[10] Klüppelberg, C, Asymptotic ordering of distribution functions and convolution semigroups, (), 77-92 · Zbl 0687.60018
[11] Leslie, J.R, On the non-closure under convolution of the subexponential family, J. appl. probab., 26, 58-66, (1989) · Zbl 0672.60027
[12] Villasenor, J.A, Further results on subexponential distributions, (1987), preprint
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