A direct proof of a fundamental theorem of renewal theory

Abstract

Let {X _n } be a sequence of independent absolutely continuous random variables not necessarily non-negative. Let h_n (χ) be the frequency function of X ₁ + ··· + X _n . A theorem is proved concerning the limiting behaviour as X ₁ tends to infinity of Σa _n h _n (χ), for a general class of sequences {a _n }. We first give a simple direct proof of a theorem due to Feller and to Täcklind that when the X _n are identically distributed with mean μ, Σ h ₂ (χ) = 1/ _μ . Less is assumed than in Feller's proof and in certain respects our conditions are much weaker than Täcklind's. Applications are given to a problem in genetics and to Tauberian theory.