The following elegant one-sided large deviation result is given by S. V. Nagaev in [2].

Theorem 0. Suppose that {Sn,n ≤ 0} is a random walk whose increments Xi are independent copies of X, where(X) = 0 and

Pr{X > x} ̃ x−αL(x) as x→ + ∞,

and where 1 < α < ∞ and L is slowly varying at ∞. Then for any ε > 0 and uniformly in x ≥ εn

Pr{Sn > x} ̃ n Pr{X > x} as n→∞.

It is the purpose of this note to point out that for lattice-valued random walks there is an analogous local limit theorem.