Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-29T02:02:31.785Z Has data issue: false hasContentIssue false

Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space

Published online by Cambridge University Press:  14 July 2016

K. J. E. Carpio*
Affiliation:
The Australian National University
D. J. Daley*
Affiliation:
The Australian National University
*
Postal address: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia.
Postal address: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia.
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.

When {Xn} is an irreducible, stationary, aperiodic Markov chain on the countable state space X = {i, j,…}, the study of long-range dependence of any square integrable functional {Yn} := {yXn} of the chain, for any real-valued function {yi: iX}, involves in an essential manner the functions Qijn = ∑r=1n(pijr − πj), where pijr = P{Xr = j | X0 = i} is the r-step transition probability for the chain and {πi: iX} = P{Xn = i} is the stationary distribution for {Xn}. The simplest functional arises when Yn is the indicator sequence for visits to some particular state i, Ini = I{Xn=i} say, in which case limsupn→∞n−1var(Y1 + ∙ ∙ ∙ + Yn) = limsupn→∞n−1 var(Ni(0, n]) = ∞ if and only if the generic return time random variable Tii for the chain to return to state i starting from i has infinite second moment (here, Ni(0, n] denotes the number of visits of Xr to state i in the time epochs {1,…,n}). This condition is equivalent to Qjin → ∞ for one (and then every) state j, or to E(Tjj2) = ∞ for one (and then every) state j, and when it holds, (Qijn / πj) / (Qkkn / πk) → 1 for n → ∞ for any triplet of states i, jk.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2007 

References

Carpio, K. J. E. (2006). Long-range dependence of Markov chains. , The Australian National University.Google Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. (1968). Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305317.Google Scholar
Daley, D. J. (1978). Upper bounds for the renewal function via Fourier methods. Ann. Prob. 6, 876884.CrossRefGoogle Scholar
Daley, D. J. (1999). The Hurst index of long-range dependent renewal processes. Ann. Prob. 27, 20352041.CrossRefGoogle Scholar
Daley, D. J. (2001). The moment index of minima. In Probability, Statistics and Seismology (J. Appl. Prob. Spec. Vol. 38A), Applied Probability Trust, Sheffield, pp. 3336.Google Scholar
Daley, D. J. (2007). Long-range dependence in a Cox process directed by an alternating renewal process. Submitted.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, 2nd edn., Springer, New York.Google Scholar
Daley, D. J. and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Process. Appl. 70, 265282.Google Scholar
Daley, D. J., Rolski, T. and Vesilo, R. (2007). Long-range dependence in a Cox process directed by a Markov renewal process. To appear in J. Appl. Math. Decision Sci.CrossRefGoogle Scholar
Feller, W. (1966). On the Fourier representation for Markov chains and the strong ratio theorem. J. Math. Mech. 15, 274283.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Heyde, C. C. (1988). Asymptotic efficiency results for the method of moments with application to estimation for queueing processes. In Queueing Theory and Its Applications, eds Boxma, O. J. and Syski, R., North-Holland, Amsterdam, pp. 405412.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Orey, S. (1961). Sums arising in the theory of Markov chains. Proc. Amer. Math. Soc. 12, 847856.Google Scholar
Scheller-Wolf, A. (2003). Necessary and sufficient conditions for delay moments in FIFO multiserver queues with an application comparing s slow servers with one fast one. Operat. Res. 51, 748758.CrossRefGoogle Scholar
Sgibnev, M. S. (1981). Renewal theorem in the case of infinite variance. Siberian Math. J. 22, 787796.Google Scholar
Sgibnev, M. S. (1996). An infinite variance solidarity theorem for Markov renewal functions. J. Appl. Prob. 33, 434438.Google Scholar