Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strong renewal theorems with infinite mean

Author: K. Bruce Erickson
Journal: Trans. Amer. Math. Soc. 151 (1970), 263-291
MSC: Primary 60.70
MathSciNet review: 0268976
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Abstract: Let F be a nonarithmetic probability distribution on $ (0,\infty )$ and suppose $ 1 - F(t)$ is regularly varying at $ \infty $ with exponent $ \alpha ,0 < \alpha \leqq 1$. Let $ U(t) = \Sigma {F^{{n^ \ast }}}(t)$ be the renewal function. In this paper we first derive various asymptotic expressions for the quantity $ U(t + h) - U(t)$ as $ t \to \infty ,h > 0$ fixed. Next we derive asymptotic relations for the convolution $ {U^ \ast }z(t),t \to \infty $, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function $ m(t) = \smallint _0^t[1 - F(x)]dx$, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case $ \alpha = 1$ to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.

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Keywords: Probability distributions, renewal function, nonarithmetic, regular and slow variation, strong renewal theorem, infinite mean, convolution, waiting times, renewal process, characteristic function, inversion formulas, weak convergence of measures, domain of attraction, local limit theorems, Karamata Tauberian theorem
Article copyright: © Copyright 1970 American Mathematical Society