On a new class of tempered stable distributions: moments and regular variation.(English)Zbl 1269.60018

$$p$$-tempered $$\alpha$$-stable distributions on $$\mathbb{R}^d$$ are infinitely divisible laws $$\mu$$ with Lévy measure $$B\mapsto M(B):= \int_{\mathbb{R}^d} 1_B(r u)q(r^p,u) r^{-\alpha-1} dr d\sigma(u)$$ ($$B$$ denoting a Borel set in $$\mathbb{R}^d \backslash\{0\}$$)), where $$\sigma \in M^b_+(S^{d-1})$$ and $$q: \mathbb{R}_+^\times \times S^{d-1}\to\mathbb{R}_+^\times$$ is a Borel function such that, for all $$u$$, $$r\mapsto q(r,u)$$ is completely monotone with $$\lim_{r\to\infty}q(r,u)=0$$. Furthermore, it is supposed that $$\alpha < 2$$, $$p>0$$ and $$\mu$$ has no Gaussian component. $$\mu$$ is called proper if $$\lim_{r\to 0}q(r,u)=1$$ for all $$u$$. If $$0<\alpha<2$$ and $$p=1$$, the class coincides with tempered $$\alpha$$-stable distributions investigated by J. Rosiński [Stochastic Processes Appl. 117, No. 6, 677–707 (2007; Zbl 1118.60037)], more general, for $$p>0$$, cf. by J. Rosiński and J. Sinclair [Banach Center Publications 90, 153-170 (2010; Zbl 1210.60048)]. In this case, $$M$$ is absolutely continuous with respect to the Lévy measure of a stable distribution.
The author generalizes the concept of Rosiński (and various others), admitting also negative values $$\alpha$$. The absolute monotone functions $$q(\cdot,u)$$ define (via Bernstein’s theorem) a family $$(Q_u)$$ of Borel measures and a re-parametrization (defined in [Rosiński, loc. cit.]) yields a unique measure $$R$$ (called here Rosiński measure). $$(Q_u)$$, $$\sigma$$ and the Lévy measure $$M$$ are determined by $$R$$ (Theorem 1). Hence, the notation $$\mu\in TS_\alpha^p(R,b)$$ is justified, where $$b$$ denotes the drift term in the Lévy-Khinchin representation of $$\mu$$.
For non-proper $$p$$-tempered $$\alpha$$-stable laws $$\mu$$, the parameters $$R$$ and $$\alpha$$ are in general not identifiable. In particular, for $$0<\alpha<2$$ and $$\beta \in (\alpha, 2)$$, $$\mu\in TS_\beta^p(R,b)$$ is representable as $$\mu\in TS_\alpha^p(R',b)$$ (for a Rosiński measure $$R'$$).
Section 3 is concerned with the existence of moments and exponential moments of $$\mu$$ and Section 4 with regular variation of the tail of a Rosiński measure $$R$$, thus determining $$\gamma$$-stable laws $$\nu$$ such that $$\mu\in TS_\alpha^p(R,b)$$ belongs to the domain of attraction of $$\nu$$.

MSC:

 60E07 Infinitely divisible distributions; stable distributions 60G51 Processes with independent increments; Lévy processes

Citations:

Zbl 1118.60037; Zbl 1210.60048
Full Text:

References:

 [1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables , 9th edn. Dover Publications, New York. · Zbl 0543.33001 [2] Allen, O. O. (1992). Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2, 951-972. · Zbl 0762.62031 [3] Aoyama, T., Maejima, M. and Rosiński, J. (2008). A subclass of type $$G$$ selfdecomposable distributions on $$\mathbb{R}^d$$. J. Theoret. Prob. 21, 14-34. · Zbl 1146.60013 [4] 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. · Zbl 1102.60013 [5] Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908-920. · Zbl 1070.60011 [6] 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. · Zbl 1215.60013 [7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27 ). Cambridge University Press. · Zbl 0617.26001 [8] 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. [9] 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. [10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol. II, 2nd edn. John Wiley, New York. · Zbl 0219.60003 [11] Grabchak, M. and Samorodnitsky, G. (2010). Do financial returns have finite or infinite variance? A paradox and an explanation. Quant. Finance 10, 883-893. · Zbl 1202.91333 [12] 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. · Zbl 1224.60023 [13] 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. [14] Hult, H. and Lindskog, F. (2006). On regular variation for infinitely divisible random vectors and additive processes. Adv. Appl. Prob. 38, 134-148. · Zbl 1106.60046 [15] 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. [16] Maejima, M. and Nakahara, G. (2009). A note on new classes of infinitely divisible distributions on $$\mathbb R^d$$. Electron. Commun. Prob. 14, 358-371. · Zbl 1189.60037 [17] Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors . John Wiley, New York. · Zbl 0990.60003 [18] Meerschaert, M. M., Zhang, Y. and Baeumer, B. (2008). Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35 , 5pp. [19] 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. · Zbl 1409.62225 [20] Rosiński, J. (2007). Tempering stable processes. Stoch. Process. Appl. 117, 677-707. · Zbl 1118.60037 [21] 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. · Zbl 1210.60048 [22] 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. [23] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman & Hall, New York. · Zbl 0925.60027 [24] 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. · Zbl 1192.60038 [25] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge University Press. · Zbl 0973.60001 [26] Terdik, G. and Woyczyński, W. A. (2006). Rosiński measures for tempered stable and related Ornstein-Uhlenbeck processes. Prob. Math. Statist. 26, 213-243. · Zbl 1134.60014 [27] Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability . VSP, Utrecht. · Zbl 0944.60006
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