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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 {X n } 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 {Y n } := {y X n } of the chain, for any real-valued function {y i : i ∈ X }, involves in an essential manner the functions Q ij n = ∑r=1 n (p ij r − πj ), where p ij r = P{X r = j | X 0 = i} is the r-step transition probability for the chain and {πi : i ∈ X } = P{X n = i} is the stationary distribution for {X n }. The simplest functional arises when Y n is the indicator sequence for visits to some particular state i, I ni = I {X n=i} say, in which case limsupn→∞ n −1var(Y 1 + ∙ ∙ ∙ + Y n ) = limsupn→∞ n −1 var(N i (0, n]) = ∞ if and only if the generic return time random variable T ii for the chain to return to state i starting from i has infinite second moment (here, N i (0, n] denotes the number of visits of X r to state i in the time epochs {1,…,n}). This condition is equivalent to Q ji n → ∞ for one (and then every) state j, or to E(T jj 2) = ∞ for one (and then every) state j, and when it holds, (Q ij n / πj ) / (Q kk n / 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)

Information

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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