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A renewal theorem in the infinite mean case. (English) Zbl 0614.60084

Let F be a distribution function generating a renewal function \(U=\sum F^{(n)}\). The key renewal theorem evaluates the asymptotic behaviour of Q*U(t) as \(t\to \infty\). The paper considers the case where \(1- F(x)=x^{-\alpha}L_ 1(x)\) \((1/2<\alpha \leq 1)\), \(Q(x)=x^{-\beta}L_ 2(x)\) (0\(\leq \beta <1)\) non-increasing and \(L_ i\) \((i=1,2)\) slowly varying. Using results of K. B. Erickson [Trans. Am. Math. Soc. 151, 263-291 (1970; Zbl 0212.516)] and N. R. Mohan [Ann. Probab. 4, 863-868 (1976; Zbl 0352.60062)] the authors show that \[ Q*U(t)\sim c(\int^{t}_{0}Q(u)du)(\int^{t}_{0}[1-F(u)]du)^{-1} \] where \(c=c(\alpha,\beta)\) a constant. The result corrects an earlier but incomplete version due to the reviewer [Ann. Math. Stat. 39, 1210-1219 (1968; Zbl 0164.191)].
The authors also provide applications of the result to second order behaviour of U(t).
Reviewer: J.L.Teugels

MSC:

60K05 Renewal theory
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