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Convolution equivalence and infinite divisibility: corrections and corollaries. (English) Zbl 1132.60015
Summary: Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.

MSC:
60E07 Infinitely divisible distributions; stable distributions
60F99 Limit theorems in probability theory
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[1] Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1 , 663–673. · Zbl 0387.60097
[2] Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43 , 347–365. (Correction: 48 , (1990), 152–153.) · Zbl 0633.60021
[3] Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13 , 263–278. · Zbl 0487.60016
[4] Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Prob. 35 , 366–383 · Zbl 1129.60014
[5] Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails , eds R. Adler, R. Feldman and M. S. Taqqu, Birkhäuser, Boston, MA, pp. 435–459. · Zbl 0923.62021
[6] Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82 , 259–269. · Zbl 0687.60017
[7] Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41 , 407–424. · Zbl 1051.60019
[8] Rogozin, B. A. (2000). On the constant in the definition of subexponential distributions. Theory Prob. Appl. 44 , 409–412. · Zbl 0971.60009
[9] Rogozin, B. A. and Sgibnev, M. S. (1999). Strongly exponential distributions, and Banach algebras of measures. Siberian Math. J. 40 , 963–971. · Zbl 0944.60037
[10] Shimura, T. and Watanabe, T. (2005). Infinite divisibility and generalized subexponentiality. Bernoulli 11 , 445–469. · Zbl 1081.60016
[11] Wang, Y., Yang, Y., Wang, K. and Cheng, D. (2007). Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications. Insurance Math. Econom. 40 , 256–266. · Zbl 1120.60033
[12] Watanabe, T. (2007). Convolution equivalence and distributions of random sums of IID. Submitted. · Zbl 1146.60014
[13] Willekens, E. (1987). Subexponentiality on the real line. Res. Rep., Katholieke Universiteit Leuven. · Zbl 0633.60025
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