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

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

K. J. E. Carpio  and
D. J. Daley
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.
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Abstract

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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: i ∈ X}, 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: i ∈ X} = 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.

Keywords

Markov chainlong-range dependenceHurst indexmoment indexfunctional

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1047 - 1055
Copyright
Copyright © Applied Probability Trust 2007 

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