A large deviation local limit theorem

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Abstract
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 andPr{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 ≥ εnPr{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.

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