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On modes of long-range dependence

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
Australian National University and Columbia University
*
∗ Postal address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
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Abstract

This paper aims at enhancing the understanding of long-range dependence (LRD) by focusing on mechanisms for generating this dependence, namely persistence of signs and/or persistence of magnitudes beyond what can be expected under weak dependence. These concepts are illustrated through a discussion of fractional Brownian noise of index H ∈ (0,1) and it is shown that LRD in signs holds if and only if ½ < H < 1 and LRD in magnitudes if and only if ¾ ≤ H < 1. An application to discrimination between two risky asset finance models, the FATGBM model of Heyde and the multifractal model of Mandelbrot, is given to illustrate the use of the ideas.

Keywords

Long-range dependencepersistence of signspersistence of magnitudesfractional Brownian motionrisky asset modelsFATGBM modelmultifractal model

MSC classification

Primary: 60G10: Stationary processes 91B70: Stochastic models
Secondary: 60G18: Self-similar processes 60G15: Gaussian processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 882 - 888
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
[2]. Cox, D. R. (1984). Long-range dependence: a review. In Statistics: An Appraisal, eds David, H. A. and David, H. T., Iowa State University Press, Ames, pp. 55–74.Google Scholar
[3]. Fisher, A., Calvet, L., and Mandelbrot, B. B. (1997). Multifractality of the Deutchmark/US Dollar exchange rates. Cowles Foundation Discussion Paper No. 1165.Google Scholar
[4]. Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234–1239.CrossRefGoogle Scholar
[5]. Heyde, C. C., and Liu, S. (2001). Empirical realities for a minimal description risky asset model. The need for fractal features. J. Korean Math. Soc. 38, 1047–1059.Google Scholar
[6]. Heyde, C. C., and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.CrossRefGoogle Scholar
[7]. Johnson, N. L., and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. John Wiley, New York.Google Scholar
[8]. Mandelbrot, B. B. (2001). Scaling in financial prices: I. Tails and dependence. Quant. Finance 1, 113–123.Google Scholar
[9]. Mandelbrot, B. B. (2001). Scaling in financial prices: II. Multifractals and the star equation. Quant. Finance 1, 124–130.CrossRefGoogle Scholar
[10]. Mandelbrot, B. B. (2001). Scaling in financial prices: III. Cartoons Brownian motions in multifractal time. Quant. Finance 1, 427–440.Google Scholar
[11]. Mandelbrot, B. B. (2001). Scaling in financial prices: IV. Multifractal concentration. Quant. Finance 1, 641–649.Google Scholar
[12]. Mandelbrot, B. B., Fisher, A., and Calvet, L. (1997). A multifractal model of asset returns. Cowles Foundation Discussion Paper No. 1164.Google Scholar
[13]. Rachev, S., and Mittnik, S. (2000). Stable Paretian Models in Finance. John Wiley, New York.Google Scholar