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Convolution equivalence and infinite divisibility

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: School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email address: [email protected]
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Abstract

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Keywords

Subexponential distributionconvolution equivalenceinfinite divisibilityrandom sumtail equivalence

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 407 - 424
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions. Springer, New York.CrossRefGoogle Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323–331.CrossRefGoogle Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640–648.Google Scholar
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. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.CrossRefGoogle Scholar
Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43, 347–365.Google Scholar
Embrechts, P., and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.Google Scholar
Embrechts, P., and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M., and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, eds Adler, R., Feldman, R. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 435–460.Google Scholar
Mantegna, R. N., and Stanley, H. E. (1994). Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73, 2946–2949.CrossRefGoogle ScholarPubMed
Pakes, A. G. (2003). Investigating the structure of truncated Lévy-stable laws. In Science and Statistics: A Festschrift for Terry Speed (Lecture Notes Monogr. Ser. 40), ed. Goldstein, D. R., Institute of Mathematical Statistics, Beachwood, OH, pp. 49–78.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sigman, K. (1999). Editorial introduction (Special issue on queues with heavy-tailed distributions). Queueing Systems 33, 1–3.CrossRefGoogle Scholar
Stam, A. J. (1973). Regular variation of the tail of a subordinated distribution. Adv. Appl. Prob. 5, 308–327.Google Scholar
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000–1011.CrossRefGoogle Scholar
Willekens, E. (1987). Subexponentiality on the real line. Res. Rep., Katholieke Universiteit Leuven.Google Scholar
Yakymiv, A. L. (2002). On the asymptotics of the density of an infinitely divisible distribution at infinity. Theory Prob. Appl. 47, 114–122. References for note added in proofGoogle Scholar
Cai, J., and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.CrossRefGoogle Scholar
Foss, S., and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37–53.CrossRefGoogle Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Sgibnev, M. S. (1990). Asymptotics of infinitely divisible distributions in R . Siberian Math. J. 31, 115–119.Google Scholar