Source: J. Appl. Probab.
Volume 49, Number 2
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.
Full-text: Access denied (no subscription
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Barron, A. (1985). Logically smooth density estimation. Doctoral Thesis, Department of Electrical Engineering, Stanford University.
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.
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.
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.
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Springer, New York.
Mathematical Reviews (MathSciNet): MR217872
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–236.
Fitzek, F. H. P. and Reisslein, M. (2001). MPEG-4 and H. 263 video traces for network performance. IEEE Network 15, 40–54.
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.
Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002–1024.
Kontoyiannis, I. (1997). Second-order noiseless source coding theorems. IEEE Trans. Inf. Theory 43, 1339–1341.
Mandelbrot, B. (1966). Forecasts of future prices, unbiased markets, and “martingale” models. J. Business 39, 242–255.
Mihalis, G. \et (2009). Scheduling policies for single-hop networks with heavy-tailed traffic. In Proc. 47th Annual Allerton Conf., pp. 112–120.
Oğuz, B. and Anantharam, V. (2010). Compressing a long range dependent renewal process. In IEEE Internat. Symp. on Information Theory Proc., pp. 1443–1447.
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.