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On a new Class of Tempered Stable Distributions: Moments and Regular Variation

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Michael Grabchak
Michael Grabchak*
Affiliation:
University of North Carolina at Charlotte
*
∗ Postal address: University of North Carolina at Charlotte, 376 Fretwell Hall, 9201 University City Blvd, Charlotte, NC 28223, USA. Email address: [email protected]
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Abstract

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We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

Keywords

Infinite divisibilitytempered stable distributionregular variationtail behavior

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1015 - 1035
Copyright
© Applied Probability Trust 

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 9th edn. Dover Publications, New York.Google Scholar
Allen, O. O. (1992). Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2, 951–972.Google Scholar
Aoyama, T., Maejima, M. and Rosiński, J. (2008). A subclass of type G selfdecomposable distributions on R d . J. Theoret. Prob. 21, 14–34.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Maejima, M. and Sato, K.-I. (2006). Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.CrossRefGoogle Scholar
Bianchi, M. L., Rachev, S. T., Kim, Y. S. and Fabozzi, F. J. (2011). Tempered infinitely divisible distributions and processes. Theory Prob. Appl. 55, 2–26.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.CrossRefGoogle Scholar
Bruno, R., Sorriso-Valvo, L., Carbone, V. and Bavassano, B. (2004). A possible truncated-Lévy-flight statistics recovered from interplanetary solar-wind velocity and magnetic-field fluctuations. Europhys. Lett. 66, 146–152.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Grabchak, M. and Samorodnitsky, G. (2010). Do financial returns have finite or infinite variance? A paradox and an explanation. Quant. Finance 10, 883–893.CrossRefGoogle Scholar
Gupta, A. K., Shanbhag, D. N., Nguyen, T. T. and Chen, J. T. (2009). Cumulants of infinitely divisible distibutions. Random Operators Stoch. Equat. 17, 103–124.Google Scholar
Gyires, T. and Terdik, G. (2009). Does the Internet still demonstrate fractal nature? In 8th Internat. Conf. Networks, IEEE Computer Society Press, Washington, DC, pp. 30–34.Google Scholar
Hult, H. and Lindskog, F. (2006). On regular variation for infinitely divisible random vectors and additive processes. Adv. Appl. Prob. 38, 134–148.CrossRefGoogle Scholar
Kim, Y. S., Rachev, S. T., Bianchi, M. L. and Fabozzi, F. J. (2010). Tempered stable and tempered infinitely divisible GARCH models. J. Banking Finance 34, 2096–2109.Google Scholar
Maejima, M. and Nakahara, G. (2009). A note on new classes of infinitely divisible distributions on R d . Electron. Commun. Prob. 14, 358–371.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.Google Scholar
Meerschaert, M. M., Zhang, Y. and Baeumer, B. (2008). Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, 5pp.CrossRefGoogle Scholar
Palmer, K. J., Ridout, M. S. and Morgan, B. J. T. (2008). Modelling cell generation times by using the tempered stable distribution. J. R. Statist. Soc. C 57, 379–397.CrossRefGoogle Scholar
Rosiński, J. (2007). Tempering stable processes. Stoch. Process. Appl. 117, 677–707.CrossRefGoogle Scholar
Rosiński, J. and Sinclair, J. L. (2010). Generalized tempered stable processes. In Stability in Probability (Banach Center Publ. 90), Polish Acad. Sci. Inst. Math. Warsaw, pp. 153–170.CrossRefGoogle Scholar
Rvačeva, E. L. (1962). On domains of attraction of multi-dimensional distributions. In Selected Translations in Mathematical Statistics and Probability, Vol. 2, American Mathematical Society, Providence, RI, pp. 183–205.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sapatinas, T. and Shanbhag, D. N. (2010). Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability. J. Multivariate Anal. 101, 500–511.CrossRefGoogle Scholar
Sato, K.-I. (1999). {Lévy Processes and Infinitely Divisible Distributions}. Cambridge University Press.Google Scholar
Terdik, G. and Woyczyński, W. A. (2006). Rosiński measures for tempered stable and related Ornstein–Uhlenbeck processes. Prob. Math. Statist. 26, 213–243.Google Scholar
Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability. VSP, Utrecht.CrossRefGoogle Scholar