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
− π
j
), where p
ij
r
= P{X
r
= j | X
0 = i} is the r-step transition probability for the chain and {π
i
: 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 limsup
n→∞
n
−1var(Y
1 + ∙ ∙ ∙ + Y
n
) = limsup
n→∞
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
/ π
j
) / (Q
kk
n
/ π
k
) → 1 for n → ∞ for any triplet of states i, j
k.