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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
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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 

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