For a given one-dimensional random walk {Sn} with a subexponential step-size distribution, we present a unifying theory to study the sequences {xn} for which $\mathsf{P}\{S_{n}>x\}\sim n\mathsf{P}\{S_{1}>x\}$ as n→∞ uniformly for x≥xn. We also investigate the stronger “local” analogue, $\mathsf{P}\{S_{n}\in(x,x+T]\}\sim n\mathsf{P}\{S_{1}\in(x,x+T]\}$ . Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory.
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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.