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2016 | OriginalPaper | Buchkapitel

3. Tempered Stable Distributions

verfasst von : Michael Grabchak

Erschienen in: Tempered Stable Distributions

Verlag: Springer International Publishing

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Abstract

In this chapter we formally define tempered stable distributions and discuss many of their properties.

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Vorheriges Kapitel Preliminaries

Nächstes Kapitel Limit Theorems for Tempered Stable Distributions

A general reference on completely monotone functions is [72].

The measurability of the family means that for any Borel set A the function f(u) = Q _u(A) is measurable.

This implies that X ₁ has a proper p-tempered α-stable distribution.

The only other case where reasonable representations are known is when p = 2 and α ∈ (0, 2). In this case [9] gives formulas in terms of confluent hypergeometric functions.

This means that S is the smallest closed subset of \(\mathbb{R}^{d}\) with R(S ^c ) = 0.

In light of Theorem 3.17, it is clear that this is not a necessary condition when α ≤ 0.

We can use l’Hôpital’s rule because the denominator is real. However, in general, l’Hôpital’s rule may fail for complex valued functions of real numbers, see [18].

M. Abramowitz and I. A. Stegun (1972). Handbook of Mathematical Functions 10th ed. Dover Publications, New York.MATH

O. E. Barndorff-Nielsen, M. Maejima, and K. Sato (2006). Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli, 12(1):1–33.MathSciNetMATH

M. L. Bianchi, S. T. Rachev, Y. S. Kim, and F. J. Fabozzi (2011). Tempered infinitely divisible distributions and processes. Theory of Probability and Its Applications, 55(1):2–26.MathSciNetCrossRefMATH

10.

P. Billingsley (1995). Probability and Measure, 3rd Ed. Wiley, New York.MATH

12.

R. M. Blumenthal and R. K. Getoor (1961). Sample functions of stochastic processes with stationary independent increments. Journal of Mathematics and Mechanics, 10:493–516.MathSciNetMATH

18.

D. S. Carter (1958). L’Hospital’s rule for complex-valued functions. The American Mathematical Monthly, 65(4), 264–266.MathSciNetCrossRefMATH

22.

R. M. Dudley and R. Norvaiša (1998). An Introduction to p-Variation and Young Integrals with emphasis on sample functions of stochastic processes. Number 1 in MaPhySto Lecture Notes. Center for Mathematical Physics and Stochastics.

23.

W. Feller (1971). An Introduction to Probability Theory and Its Applications Volume II, 2nd Ed. John Wiley & Sons, Inc., New York.MATH

24.

B. Fristedt and S. J. Taylor (1973). Strong variation for the sample functions of a stable process. Duke Mathematical Journal, 40(2):259–278.MathSciNetCrossRefMATH

27.

M. Grabchak (2012). On a new class of tempered stable distributions: Moments and regular variation. Journal of Applied Probability, 49(4):1015–1035.MathSciNetCrossRefMATH

51.

M. Maejima and G. Nakahara (2009). A note on new classes of infinitely divisible distributions on \(\mathbb{R}^{d}\). Electronic Communications in Probability, 14:358–371.MathSciNetCrossRefMATH

54.

M. M. Meerschaert and H. Scheffler (2001). Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. John Wiley & Sons, New York.MATH

56.

I. Monroe (1972). On the γ-variation of processes with stationary independent increments. Annals of Mathematical Statistics, 43(4):1213–1220.MathSciNetCrossRefMATH

65.

J. Rosiński (2007). Tempering stable processes. Stochastic Processes and their Applications, 117(6):677–707.MathSciNetCrossRefMATH

66.

J. Rosiński and J. L. Sinclair (2010). Generalized tempered stable processes. Banach Center Publications, 90:153–170.MathSciNetCrossRefMATH

67.

E. L. Rvačeva (1962). On domains of attraction of multi-dimensional distributions. In Selected Translations in Mathematical Statistics and Probability Vol. 2, pg. 183–205. American Mathematical Society, Providence. Translated by S. G. Ghurye.

68.

G. Samorodnitsky and M. S. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.MATH

69.

K. Sato (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.MATH

72.

R. L. Schilling, R. Song, and Z. Vondraček (2012). Bernstein Functions: Theory and Applications. DeGruyter, Berlin.CrossRefMATH

73.

P. J. Smith (1995). A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. The American Statistician, 49(2): 217–218.MathSciNet

78.

V. V. Uchaikin and V. M. Zolotarev (1999). Chance and Stability: Stable Distributions and their Applications. VSP BV, Utrecht.CrossRefMATH

Titel: Tempered Stable Distributions
verfasst von: Michael Grabchak
Verlag: Springer International Publishing
Buch: Tempered Stable Distributions
Print ISBN: 978-3-319-24925-4

Electronic ISBN: 978-3-319-24927-8

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-24927-8_3