Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T15:30:34.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Some stationary processes in discrete and continuous time

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

O. E. Barndorff-Nielsen ,
J. L. Jensen  and
M. Sørensen
O. E. Barndorff-Nielsen*
Affiliation:
University of Aarhus
J. L. Jensen*
Affiliation:
University of Aarhus
M. Sørensen*
Affiliation:
University of Copenhagen
*
∗ Postal address: Department of Theoretical Statistics, University of Aarhus, DK-8000 Aarhus C, Denmark.
∗ Postal address: Department of Theoretical Statistics, University of Aarhus, DK-8000 Aarhus C, Denmark.
∗∗ Department of Theoretical Statistics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. Email address: michael@math.ku.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

A number of stationary stochastic processes are presented with properties pertinent to modelling time series from turbulence and finance. Specifically, the one-dimensional marginal distributions have log-linear tails and the autocorrelation may have two or more time scales. Discrete time models with a given marginal distribution are constructed as sums of independent autoregressions. A similar construction is made in continuous time by considering sums of Ornstein-Uhlenbeck-type processes. To prepare for this, a new property of self-decomposable distributions is presented. Also another, rather different, construction of stationary processes with generalized logistic marginal distributions as an infinite sum of Gaussian processes is proposed. In this way processes with continuous sample paths can be constructed. Multivariate versions of the various constructions are also given.

Keywords

Autoregressiongeneralized logistic distributionLévy processesoperator self-decomposabilityOrnstein-Uhlenbeck processesturbulenceself-decomposability

MSC classification

Primary: 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 989 - 1007
Copyright
Copyright © Applied Probability Trust 1998 

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

Bagnold, R. A. and Barndorff-Nielsen, O. E. (1980). The pattern of natural size distributions. Sedimentology 27, 199–207.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (1979). Models for non-Gaussian variation with applications to turbulence. Proc. R. Soc. London A 368, 501–520.Google Scholar
Barndorff-Nielsen, O. E. (1982). The hyperbolic distribution in statistical physics. Scand. J. Statist. 9, 43–46.Google Scholar
Barndorff-Nielsen, O. E. (1998a). Probability and Statistics: selfdecomposability, finance and turbulence. In Probability towards 2000, ed. Accardi, L. and Heyde, C. C. (Lecture Notes in Statist. 128). Springer, New York, pp. 47–57.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (1998b). Processes of normal inverse Gaussian type. Finance and Stochastics 2, 41–68.Google Scholar
Barndorff-Nielsen, O. E. and Blæsild, P. (1983). Hyperbolic distributions. In Encyclopedia of Statistical Science, Vol 3. ed. Johnson, N. L. and Kotz, S.. Wiley, New York.Google Scholar
Barndorff-Nielsen, O. E. and Christiansen, C. (1988). Erosion, deposition and size distributions of sand. Proc. R. Soc. London A 417, 335–352.Google Scholar
Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1989). Wind shear and hyperbolic distributions. Boundary-Layer Meteorology 49, 417–431.Google Scholar
Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1990). Parametric modelling of turbulence. Phil. Trans. R. Soc. Lond. A 332, 439–455.Google Scholar
Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1993). A statistical model for the streamwise component of a turbulent velocity field. Ann. Geophys. 11, 99–103.Google Scholar
Barndorff-Nielsen, O. E., Kent, J. and Sørensen, M. (1982). Normal variance-mean mixtures and z-distributions. Int. Statist. Rev. 50, 145–159.Google Scholar
Blæsild, P., (1981). The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen's bean data. Biometrika 68, 251–263.CrossRefGoogle Scholar
Cox, D. R. (1981). Statistical analysis of time series: some recent developments (with discussion). Scand. J. Statist. 8, 93–115.Google Scholar
Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281–299.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Gaver, D. P. and Lewis, P. A. W. (1980). First-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1965). Tables of Integrals, Series, and Products, 4th edn. Academic Press, New York.Google Scholar
Halgreen, C. (1979). Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitsth. 47, 13–17.Google Scholar
Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.Google Scholar
Jurek, Z. J. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. Wiley, New York.Google Scholar
Jurek, Z. J. and Vervaat, W. (1983). An integral representation for self-decomposable Banach space valued random variables. Z. Wahrscheinlichkeitsth. 62, 247–262.CrossRefGoogle Scholar
Küchler, U., Neumann, K., Sørensen, M. and Streller, A. (1994). Stock returns and hyperbolic distributions. To appear in Stable Modelling in Finance, ed. Mittnik, S. and Rachev, S. T.. Discussion Paper No. 23, Sonderforschungsbereich 373, Humboldt-Universität zu Berlin.Google Scholar
Loève, M., (1955). Probability Theory. Van Nostrand, New York.Google Scholar
Lukacs, E. (1969). A characterization of stable processes. J. Appl. Prob. 6, 409–418.CrossRefGoogle Scholar
Sato, K. and Yamazato, M. (1982). Stationary processes of Ornstein–Uhlenbeck type. In Probability Theory and Mathematical Statistics%, ed. Ito, K. and Prohorov, J. V. (Lecture Notes in Math. 1021). Springer-Verlag, Berlin.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., Watanabe, T. and Yamazato, M. (1994). Recurrence conditions for multidimensional processes of Ornstein–Uhlenbeck type. J. Math. Soc. Japan 46, 245–265.CrossRefGoogle Scholar
Sim, C. H. (1993). First-order autoregressive logistic processes. J. Appl. Prob. 30, 467–470.CrossRefGoogle Scholar
Wolfe, S. J. (1982). On a continuous analogue of the stochastic difference equation X_{n}=ρ Xn-1+Bn . Stoch. Proc. Appl. 12, 301–312.Google Scholar