A strong renewal theorem for generalized renewal functions in the infinite mean case
Let F(x) be a nonarithmetic c.d.f. on (0,\(\infty)\) such that \(1- F(x)=x^{-\alpha}L(x)\), where L(x) is slowly varying and \(0\leq \alpha \leq 1\). Let a(x) be regularly varying with exponent \(\beta\) \(\geq -1\). A strong renewal theorem (of Blackwell type) for generalized renewal functions of the form G(t)\(\equiv \sum ^{\infty}_{n=0}a(n)F^ n(t)\) is proved here, thus extending recent work of \textit{P. Embrechts}, \textit{M. Maejima} and \textit{E. Omey} [Ann. Probab. 12, 561-570 (1984; Zbl 0537.60087)] and that of \textit{K. B. Erickson} [Trans. Am. Math. Soc. 151, 263-291 (1970; Zbl 0212.516)].
scientific article; zbMATH DE number 4020149