Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T08:25:44.159Z Has data issue: false hasContentIssue false
  • English
  • Français

Convolution Equivalence and Infinite Divisibility: Corrections and Corollaries

Part of: Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Subexponential distributionsconvolution equivalenceinfinite divisibilityrandom sumstail equivalence

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60F99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 295 - 305
Copyright
Copyright © Applied Probability Trust 2007 

References

Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
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.)CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.CrossRefGoogle Scholar
Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Prob. 35, 366–383 CrossRefGoogle Scholar
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails, eds Adler, R., Feldman, R. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 435–459.Google Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.CrossRefGoogle Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.Google Scholar
Rogozin, B. A. (2000). On the constant in the definition of subexponential distributions. Theory Prob. Appl. 44, 409–412.Google Scholar
Rogozin, B. A. and Sgibnev, M. S. (1999). Strongly exponential distributions, and Banach algebras of measures. Siberian Math. J. 40, 963–971.CrossRefGoogle Scholar
Shimura, T. and Watanabe, T. (2005). Infinite divisibility and generalized subexponentiality. Bernoulli 11, 445–469.Google Scholar
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.Google Scholar
Watanabe, T. (2007). Convolution equivalence and distributions of random sums of IID. Submitted.Google Scholar
Willekens, E. (1987). Subexponentiality on the real line. Res. Rep., Katholieke Universiteit Leuven.Google Scholar