A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

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.

For a random variable (RV) X its moment index κ(X) ≡ sup{κ : E(|X|κ ) < ∞} lies in 0 ≤ κ (X) ≤∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ (min(X, Y)) ≥ κ(X) + κ (Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess' RVs Xs and Ys whose distributions are the integrated tails of X and Y, κ (X) + κ (Y) ≤ κ (min(Xs, Ys )) + 2 ≤ κ (min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.

For a stationary long-range dependent point process N(.) with Palm distribution P0, the Hurst index H ≡ sup{h : lim sup t→∞t -2h var N(0,t] = ∞} is related to the moment index κ ≡ sup{k : E0(T k ) < ∞} of a generic stationary interval T between points (E0 denotes expectation with respect to P0) by 2H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than ½. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E0(T 2) < ∞.