Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T20:35:30.271Z Has data issue: false hasContentIssue false
  • English
  • Français

Hurst Index of Functions of Long-Range-Dependent Markov Chains

Part of: Markov processes Mathematical finance Computer system organization Theory of data

Published online by Cambridge University Press:  04 February 2016

Barlas Oğuz  and
Venkat Anantharam
Barlas Oğuz*
Affiliation:
University of California, Berkeley
Venkat Anantharam*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA.
∗ Postal address: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

Keywords

Markov chainlong-range dependenceHurst index

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68M20: Performance evaluation; queueing; scheduling 68P30: Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) 91G70: Statistical methods, econometrics
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 451 - 471
Copyright
© Applied Probability Trust 

References

Barron, A. (1985). Logically smooth density estimation. , Department of Electrical Engineering, Stanford University.Google Scholar
Beran, J., Sherman, R., Taqqu, M. S. and Willinger, W. (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43, 1566–1579.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Carpio, K. J. E. and Daley, D. J. (2007). Long-range dependence of Markov chains in discrete time on countable state space. J. Appl. Prob. 44, 1047–1055.CrossRefGoogle Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Springer, New York.Google Scholar
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–236.CrossRefGoogle Scholar
Fitzek, F. H. P. and Reisslein, M. (2001). MPEG-4 and H. 263 video traces for network performance. IEEE Network 15, 40–54.CrossRefGoogle Scholar
Garrett, M. W. and Willinger, W. (1994). Analysis, modeling and generation of self-similar VBR video traffic. In Proc. ACM SIGCOMM '94 (London, UK), ACM, pp. 269–280.CrossRefGoogle Scholar
Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002–1024.CrossRefGoogle Scholar
Kontoyiannis, I. (1997). Second-order noiseless source coding theorems. IEEE Trans. Inf. Theory 43, 1339–1341.CrossRefGoogle Scholar
Mandelbrot, B. (1966). Forecasts of future prices, unbiased markets, and “martingale” models. J. Business 39, 242–255.CrossRefGoogle Scholar
Mihalis, G. et al. (2009). Scheduling policies for single-hop networks with heavy-tailed traffic. In Proc. 47th Annual Allerton Conf., pp. 112–120.Google Scholar
Oğuz, B. and Anantharam, V. (2010). Compressing a long range dependent renewal process. In IEEE Internat. Symp. on Information Theory Proc., pp. 1443–1447.Google Scholar
Rose, O. (1995). Statistical properties of MPEG video traffic and their impact on traffic modeling in ATM systems. In Proc. 20th Annual IEEE Conf. on Local Computer Networks, IEEE Computer Society Press, Washington, DC, pp. 397–406.Google Scholar