A strong renewal theorem for generalized renewal functions in the infinite mean case

Abstract

Let F(x) be a nonarithmetic c.d.f. on (0, ∞) such that 1 -F(x)=x. α L(x), where L(x) is slowly varying and 0 ≤ α ≤ 1. Let a(x) be regularly varying with exponent β≥1. A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form G(t) ≡ ∑ ^∞_n=0a(n)Fⁿ(t) is proved here, thus extending the recent work of Embrechts, Maejima and Omey [1] and that of Erickson [4].