Long-range dependence of Markov chains in discrete time on countable state space

When {X_n} is an irreducible, stationary, aperiodic Markov chain on the countable state space χ = {i, j, ...}, the study of long-range dependence of any square integrable functional {Y_n} := {yX _n} of the chain, for any real-valued function {y_i : i ε χ}, involves in an essential manner the functions O_ij ⁿ = ∑_r=1ⁿ (p_ij^r - π_j), where p_ij^r = P{X_r = j | X₀ = i) is the r-step transition probability for the chain and {π_i ε χ} = 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_{Xn=i} say, in which case lim sup _n→∞ n^-1 var(Y₁ + ⋯ + Y _n) = lim sup_n→∞ n^-1 (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_jiⁿ → ∞ for one (and then every) state j, or to E(T_jj²) = ∞ for one (and then every) state j, and when it holds, (Q_ijⁿ/π_j)/ {Q_kkⁿ/π_k) → 1 for n → ∞ for any triplet of states i, j, k.