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SECOND ORDER SUBEXPONENTIALITY AND INFINITE DIVISIBILITY

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  22 June 2020

TOSHIRO WATANABE Open the ORCID record for TOSHIRO WATANABE [Opens in a new window]
TOSHIRO WATANABE*
Affiliation:
Center for Mathematical Sciences,The University of Aizu, Aizu-Wakamatsu, Fukushima965-8580, Japan
*
e-mail: [email protected]
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Abstract

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

Keywords

second order subexponentialitylocal subexponentialityinfinite divisibility

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 367 - 390
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by N. Ross

References

Asmussen, S., Foss, S. and Korshunov, D., ‘Asymptotics for sums of random variables with local subexponential behaviour’, J. Theoret. Probab. 16 (2003), 489–518.CrossRefGoogle Scholar
Baltrunas, A., ‘Second order behaviour of ruin probabilities’, Scand. Actuar. J. 2 (1999), 120–133.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular Variation (Cambridge University Press, Cambridge, 1984).Google Scholar
Bondesson, L., ‘On the Lévy measure of the lognormal and the logcauchy distributions’, Methodol. Comput. Appl. Probab. 4 (2002), 243–256.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S., ‘Functions of probability measures’, J. Anal. Math. 26 (1973), 255–302.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N., ‘Subexponentiality and infinite divisibility’, Z. Wahrscheinlichkeitstheorie verw. Geb. 49 (1979), 335–347.CrossRefGoogle Scholar
Foss, S., Korshunov, D. and Zachary, S., An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn, Springer Series in Operations Research and Financial Engineering (Springer, New York, 2013).CrossRefGoogle Scholar
Geluk, J. L., ‘Second order tail behaviour of a subordinated probability distribution’, Stochastic Process. Appl. 40 (1992), 325–337.CrossRefGoogle Scholar
Geluk, J. L. and Pakes, A. G., ‘Second order subexponential distributions’, J. Aust. Math. Soc. 51 (1991), 73–87.Google Scholar
Jiang, T., Wang, Y., Cui, Z. and Chen, Y., ‘On the almost decrease of a subexponential density’, Statist. Probab. Lett. 153 (2019), 71–79.CrossRefGoogle Scholar
Klüppelberg, C., ‘Subexponential distributions and integrated tails’, J. Appl. Probab. 25 (1988), 132–141.CrossRefGoogle Scholar
Klüppelberg, C., ‘Subexponential distributions and characterizations of related classes’, Probab. Theory Related Fields 82 (1989), 259–269.CrossRefGoogle Scholar
Lin, J., ‘Second order subexponential distributions with finite mean and their applications to subordinated distributions’, J. Theoret. Probab. 25 (2012), 834–853.CrossRefGoogle Scholar
Omey, E. and Willekens, E., ‘Second order behaviour of the tail of a subordinated probability distribution’, Stochastic Process. Appl. 21 (1986), 339–353.CrossRefGoogle Scholar
Omey, E. and Willekens, E., ‘Second-order behaviour of distributions subordinate to a distribution with finite mean’, Comm. Statist. Stochastic Models 3 (1987), 311–342.CrossRefGoogle Scholar
Pakes, A. G., ‘Convolution equivalence and infinite divisibility’, J. Appl. Probab. 41 (2004), 407–424.CrossRefGoogle Scholar
Sato, K., Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68 (Cambridge University Press, Cambridge, 2013).Google Scholar
Shimura, T. and Watanabe, T., ‘Subexponential densities of compound Poisson sums and the supremum of a random walk’, Preprint, 2020, arXiv:2001.11362.Google Scholar
Steutel, F. W. and van Harn, K., Infinite Divisibility of Probability Distributions on the Real Line, Pure and Applied Mathematics: A Series of Monographs and Textbooks, 259 (Marcel Dekker, New York–Basel, 2004).Google Scholar
Watanabe, T., ‘Convolution equivalence and distributions of random sums’, Probab. Theory Related Fields 142 (2008), 367–397.CrossRefGoogle Scholar
Watanabe, T., ‘Subexponential densities of infinitely divisible distributions on the half line’, Lith. Math. J. (2020), to appear.CrossRefGoogle Scholar
Watanabe, T. and Yamamuro, K., ‘Local subexponentiality of infinitely divisible distributions’, J. Math-for-Ind. 1 (2009), 81–90.Google Scholar
Watanabe, T. and Yamamuro, K., ‘Local subexponentiality and self-decomposability’, J. Theoret. Probab. 23 (2010), 1039–1067.CrossRefGoogle Scholar