Long-range dependence of Markov chains in discrete time on countable state space - Carpio, Kristine Joy E. (Animo Repository, 2007)
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.