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Hurst index of functions of long-range-dependent Markov chains. (English) Zbl 1266.60127

Recall that a positive recurrent, aperiodic Markov chain is said to have long-range dependence (LRD) if the indicator function of a particular state has the LRD property. It can be shown that this is equivalent to the return time distribution of that state having infinite variance. The authors study whether there are other instantaneous functions of such a Markov chain that exhibit LRD. They formulate conditions which ensure that the function of the Markov chain inherits the same degree of LRD as the underlying Markov chain. Detailed examples from queueing networks, source compression and finance are discussed.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
91G70 Statistical methods; risk measures
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References:

[1] Barron, A. (1985). Logically smooth density estimation. Doctoral Thesis, Department of Electrical Engineering, Stanford University.
[2] 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.
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation . Cambridge University Press. · Zbl 0667.26003
[4] 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. · Zbl 1139.60036
[5] Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities . Springer, New York. · Zbl 0146.38401
[6] Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223-236.
[7] Fitzek, F. H. P. and Reisslein, M. (2001). MPEG-4 and H. 263 video traces for network performance. IEEE Network 15, 40-54.
[8] 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.
[9] Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002-1024. · Zbl 1084.62516
[10] Kontoyiannis, I. (1997). Second-order noiseless source coding theorems. IEEE Trans. Inf. Theory 43, 1339-1341. · Zbl 0878.94035
[11] Mandelbrot, B. (1966). Forecasts of future prices, unbiased markets, and “martingale” models. J. Business 39, 242-255.
[12] Mihalis, G. \et (2009). Scheduling policies for single-hop networks with heavy-tailed traffic. In Proc. 47th Annual Allerton Conf. , pp. 112-120.
[13] Oğuz, B. and Anantharam, V. (2010). Compressing a long range dependent renewal process. In IEEE Internat. Symp. on Information Theory Proc. , pp. 1443-1447.
[14] 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.
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