Abstract
Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.
AMS 1991 subject classifications: Primary 60G10; 60G18 secondary 62M10
Received 5 November 1996; revision received 11 March 1997.
Chapter PDF
Similar content being viewed by others
Keywords
References
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.
Cox, D. R. (1984) Long-range dependence: a review. In Statistics: an Appraisal, ed. H. A. David and H. T. David. Iowa State University Press, Ames. pp. 55–74.
Heyde, C. C. and Dai, W. (1996) On the robustness to small trends of estimation based on the smoothed periodogram. J. Time Series Anal 17, 141–150.
Loeve, M. (1963) Probability Theory. 3rd edn. Van Nostrand, Princeton, NJ.
Mittnik, S. and Rachev, S. T. (1997) Modeling Financial Assets with Alternative Stable Models. Wiley, New York.
Painter, S. (1995) Random fractal models of heterogeneity: the Lévy-stable approach. Math. Geol. 27, 813–830.
Painter, S. (1996) Existence of non-Gaussian scaling behaviour in heterogeneous sedimentary formations. Water Resources Res. 32, 1183–1195.
Painter, S. (1996) Stochastic interpolation of aquifer properties using fractional Lévy motion. Water Resouces Res. 32, 1323–1332.
Peters, E. E. (1991) Chaos and Order in the Capital Markets. Wiley, New York.
Samorodnitsky, G. and Taqqu, M. S. (1994) Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, New York.
Turcotte, D. L. (1994) Fractal theory and the estimation of extreme floods. J. Res. Nat. Inst. Stand. Tech. 99, 377–389.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer New York
About this chapter
Cite this chapter
Heyde, C.C., Yang, Y. (2010). On Defining Long-Range Dependence. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_54
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5823-5_54
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-5822-8
Online ISBN: 978-1-4419-5823-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)