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January, 1987 A Renewal Theorem in the Infinite Mean Case
Kevin K. Anderson, Krishna B. Athreya
Ann. Probab. 15(1): 388-393 (January, 1987). DOI: 10.1214/aop/1176992277


Let $F(\cdot)$ be a c.d.f. on $(0, \infty)$ such that $1 - F(x)$ is regularly varying with exponent $-\alpha, \frac{1}{2} < \alpha \leq 1$. Let $Q(\cdot): \mathscr{R}^+ \rightarrow \mathscr{R}^+$ be nonincreasing and regularly varying with exponent $-\beta, 0 \leq \beta < 1$. Then, as $t \rightarrow \infty, (U \ast Q)(t) \equiv \int_{\lbrack 0,t\rbrack}Q(t - u)U(du)$ is asymptotic to $c(\alpha, \beta)(\int^t_0Q(u) du)(\int^t_0(1 - F(u)) du)^{-1}$, where $U(\cdot)$ is the renewal function associated with $F(\cdot)$ and $c(\alpha, \beta)$ is a suitable constant. This is an improved version of a theorem due to Teugels, whose proof appears to be incomplete. Applications of the result to the second order behavior of $U(t)$ in some special cases are also given.


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Kevin K. Anderson. Krishna B. Athreya. "A Renewal Theorem in the Infinite Mean Case." Ann. Probab. 15 (1) 388 - 393, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0614.60084
MathSciNet: MR877611
Digital Object Identifier: 10.1214/aop/1176992277

Primary: 60K05

Keywords: key renewal theorem , regular variation , Renewal function , second order behavior

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 1 • January, 1987
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