Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:01:37.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate subordination, self-decomposability and stability

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Ole E. Barndorff-Nielsen ,
Jan Pedersen  and
Ken-Iti Sato
Ole E. Barndorff-Nielsen*
Affiliation:
MaPhySto and University of Aarhus
Jan Pedersen*
Affiliation:
University of Aarhus
Ken-Iti Sato*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗ Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗∗ Email address: oebn@imf.au.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

Keywords

Levy processesmultiparameter Levy processesmultivariate subordinationoperator self-decomposabilityoperator stability

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 160 - 187
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barndorff-Nielsen, O. E. (1980). Conditionality resolutions. Biometrika 67, 293–310.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Blæsild, P. (1988). Combination of reproductive models. Ann. Statist. 16, 323–341.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. (1999). Subordinators: examples and applications. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1717), ed. Bernard, P.. Springer, Berlin.CrossRefGoogle Scholar
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, Berlin.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Griffiths, R. C. (1984). Characterization of infinitely divisible multivariate gamma distributions. J. Multivar. Anal. 15, 13–20.Google Scholar
Halgreen, C. (1979). Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitsth. 47, 13–17.Google Scholar
Ismail, M. E. H. and Kelker, D. H. (1979). Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10, 884–901.Google Scholar
Jurek, Z. J. (1983). Limit distributions and one-parameter groups of linear operators on Banach spaces. J. Multivar. Anal. 13, 578–604.CrossRefGoogle Scholar
Jurek, Z. J. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.Google Scholar
Loève, M., (1977). Probability Theory, Vol. 1, 4th edn. Springer, New York.Google Scholar
Loève, M., (1978). Probability Theory, Vol. 2, 4th edn. Springer, New York.Google Scholar
Maejima, M., Sato, K. and Watanabe, T. (1999). Operator semi-selfdecomposability, (C,Q)-decomposability and related nested classes. Tokyo J. Math. 22, 473–509.Google Scholar
Moran, P. A. P. and Vere-Jones, D. (1969). The infinite divisibility of multivariate gamma distributions. Sankhyā A 32, 191–194.Google Scholar
Sato, K. (1980). Class L of multivariate distributions and its subclasses. J. Multivar. Anal. 10, 207–232.Google Scholar
Sato, K. (1987). Strictly operator stable distributions. J. Multivar. Anal. 22, 278–295.CrossRefGoogle Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sato, K. and Yamazato, M. (1983). Stationary processes of Ornstein–Uhlenbeck type. In Probability Theory and Mathematical Statistics (Lecture Notes Math. 1021), ed. Itô, K. and Prokhorov, J. V.. Springer, Berlin, pp. 278–295.Google Scholar
Sato, K. and Yamazato, M. (1984). Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stoch. Proc. Appl. 17, 73–100.Google Scholar
Sato, K. and Yamazato, M. (1985). Completely operator-selfdecomposable distributions and operator-stable distributions. Nagoya Math. J. 97, 71–94.CrossRefGoogle Scholar
Sharpe, M. (1969). Operator-stable probability distributions on vector groups. Trans. Amer. Math. Soc. 136, 51–65.CrossRefGoogle Scholar
Skorohod, A. V. (1991). Random Processes with Independent Increments. Kluwer Academic Press, Dordrecht.CrossRefGoogle Scholar
Urbanik, K. (1972a). Lévy's probability measures on Euclidean spaces. Studia Math. 44, 119–148.Google Scholar
Urbanik, K. (1972b). Slowly varying sequences of random variables. Bull. Acad. Polon. Sci. Sér. Sci. Phys. Astronom. 20, 679–682.Google Scholar
Urbanik, K. (1973). Limit laws for sequences of normed sums of random variables satisfying some stability conditions. In Multivariate Analysis III, ed. Krishnaiah, P. R.. Academic Press, New York, pp. 225–237.Google Scholar
Vere-Jones, D. (1967). The infinite divisibility of bivariate gamma distributions. Sankhyā A 29, 13–20.Google Scholar
Wolfe, S. J. (1982). On a continuous analogue to the stochastic difference equation X_n=ρ X_n-1+B_n. Stoch. Proc. Appl. 12, 301–312.Google Scholar