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Multivariate self-similar processes: second-order theory

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

M. Bladt
M. Bladt*
Affiliation:
University of Aalborg
*
∗ Postal address: Department of Mathematics and Computer Science, University of Aalborg, Fredrik Bajers Vej 7, DK-9220 Aalborg 0, Denmark.
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Abstract

The aim of this paper is to extend the existing theory of second-order self-similar processes as defined by Cox (1984) from the univariate case to higher dimensions. Multivariate self-similar processes defined in terms of second-order theory for stationary time series can be used as models for long-range dependent observations when the marginal observations are long-range dependent. An interesting question concerns the correlation structure within the processes when the marginal processes are correlated. We show that the self-similarity requirement, as defined in this article, implies a cross-correlation structure similar to that for the marginal processes. This occurs both in the time domain and in the frequency domain. This fact can be used to obtain generalized least squares estimates for the long-range dependence parameters. We discuss some difficulties concerning estimation based on simulations.

Keywords

SECOND-ORDER STATIONARY TIME SERIESLONG-RANGE DEPENDENT OBSERVATIONS

MSC classification

Primary: 60G18: Self-similar processes
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 139 - 147
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Bladt, M. Multivariate Second Order Self-Similar Processes. Master's thesis, University of Aarhus.CrossRefGoogle Scholar
[2] Cox, D. R. (1984) Long range dependence. In A Review in Statistics: An Appraisal, pp. 55–74. Iowa State University.Google Scholar
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