Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X1+⋯+XnSn=X1+⋯+Xn be a random walk, where the steps XnXn are independent random variables having a finite number of possible distributions, and consider general series of the formequation(∗)∑n⩾1anP(|Sn|⩾εbn),ε>0, with an⩾0,∑n⩾1an=∞ and bn↗∞. Under mild auxiliary assumptions on the sequences (an)n⩾1(an)n⩾1 and (bn)n⩾1(bn)n⩾1, we give necessary and sufficient conditions for the convergence of the series (∗) for any ε>0ε>0.