Series of tail distributions of finitely inhomogeneous random walks

Let S n = X 1 + ⋯ + X n be a random walk, where the steps X n are independent random variables having a finite number of possible distributions, and consider general series of the form(∗) ∑ n ⩾ 1 a n P ( | S n | ⩾ ε b n ) , ε > 0 , with a n ⩾ 0 , ∑ n ⩾ 1 a n = ∞ and b n ↗ ∞ . Under mild auxiliary assumptions on the sequences ( a n ) n ⩾ 1 and ( b n ) n ⩾ 1 , we give necessary and sufficient conditions for the convergence of the series (∗) for any ε > 0 .