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A NOTE ON THE CLOSURE OF CONVOLUTION POWER MIXTURES (RANDOM SUMS) OF EXPONENTIAL DISTRIBUTIONS

Part of: Distribution theory - Probability Limit theorems Distribution theory

Published online by Cambridge University Press:  01 February 2008

J. M. P. ALBIN
J. M. P. ALBIN*
Affiliation:
Department of Mathematics, Chalmers University of Technology, 412 96 Gothenburg, Sweden (email: palbin@math.chalmers.se)
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Abstract

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We make a correction to an important result by Cline [D. B. H. Cline, ‘Convolutions of distributions with exponential tails’, J. Austral. Math. Soc. (Series A)43 (1987), 347–365; D. B. H. Cline, ‘Convolutions of distributions with exponential tails: corrigendum’, J. Austral. Math. Soc. (Series A)48 (1990), 152–153] on the closure of the exponential class under convolution power mixtures (random summation).

Keywords

compound Poisson distributionconvolution powerexponential distributionrandom sumTauberian theorem

MSC classification

Secondary: 60F99: None of the above, but in this section 62E20: Asymptotic distribution theory 60E07: Infinitely divisible distributions; stable distributions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 1 - 7
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Albin, J. M. P. and Sundén, M., ‘On the asymptotic behaviour of Lévy processes. Part I: subexponential and exponential processes’, Preprint,2007. http://www.math.chalmers.se/∼palbin/levyI.pdf.Google Scholar
[2]Braverman, M., ‘Suprema and sojourn times of Lévy processes with exponential tails’, Stochastic Process. Appl. 68 (1997), 265–283.CrossRefGoogle Scholar
[3]Braverman, M. and Samorodnitsky, G., ‘Functionals of infinitely divisible stochastic processes with exponential tails’, Stochastic Process. Appl. 56 (1995), 207–231.CrossRefGoogle Scholar
[4]Cline, D. B. H., ‘Convolutions of distributions with exponential tails’, J. Austral. Math. Soc. (Series A) 43 (1987), 347–365.CrossRefGoogle Scholar
[5]Cline, D. B. H., ‘Convolutions of distributions with exponential tails: corrigendum’, J. Austral. Math. Soc. (Series A) 48 (1990), 152–153.CrossRefGoogle Scholar
[6]Embrechts, P. and Goldie, M., ‘On closure and factorization properties of subexponential and related distributions’, J. Austral. Math. Soc. (Series A) 29 (1980), 243–256.CrossRefGoogle Scholar
[7]Embrechts, P. and Goldie, M., ‘On convolution tails’, Stochastic Process. Appl. 13 (1982), 263–278.CrossRefGoogle Scholar
[8]Embrechts, P., Goldie, M. and Veraverbeke, N., ‘Subexponentiality and infinite divisibility’, Z. Wahrsch. 49 (1979), 335–347.CrossRefGoogle Scholar
[9]Pakes, A. G., ‘Convolution equivalence and infinite divisibility’, J. Appl. Probab. 44 (2007), 295–305.CrossRefGoogle Scholar
[10]Pitman, E. J. G., ‘Subexponential distribution functions’, J. Austral. Math. Soc. (Series A) 29 (1980), 337–347.CrossRefGoogle Scholar
[11]Sato, K., Lévy processes and infinitely divisible distributions (Cambridge University Press, Cambridge, 1999).Google Scholar
[12]Sgibner, M. S., ‘Asymptotics of inifinitely divisible distribution in ’, Siberian Math. J. 31 (1990), 115–119.CrossRefGoogle Scholar
[13]Shimura, T. and Watanabe, T., ‘Infinite divisibility and generalized subexponentiality’, Bernoulli 11 (2005), 445–469.CrossRefGoogle Scholar