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For a given one-dimensional random walk {S_n} with a subexponential step-size distribution, we present a unifying theory to study the sequences {x_n} for which as n→∞ uniformly for x≥x_n. We also investigate the stronger “local” analogue, . Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory.

When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.