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September, 1964 Present Value of a Renewal Process
Giorgio Dall'Aglio
Ann. Math. Statist. 35(3): 1326-1331 (September, 1964). DOI: 10.1214/aoms/1177703289


This paper studies the present cost $C(\rho)$ of a renewal process, defined as the sum of the values of the costs of the replacements, considered at the starting time of the renewal process, with a compound interest. The characteristic function of $C(\rho)$ is found when the inter-arrival times $X_j$ are negatively exponentially distributed. The asymptotic properties of $C(\rho)$ as the force of interest $\rho$ tends to zero are studied in the general case, obtaining that if $X_1$ has finite moments of all orders, $C(\rho)$ is asymptotically normal; moreover, if we assume $EX^2_1 < \infty$, then the existence of all the moments of $X_1$ is necessary in order that the moments of $C(\rho)$ converge to the moments of the normal distribution.


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Giorgio Dall'Aglio. "Present Value of a Renewal Process." Ann. Math. Statist. 35 (3) 1326 - 1331, September, 1964.


Published: September, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0129.10601
MathSciNet: MR164383
Digital Object Identifier: 10.1214/aoms/1177703289

Rights: Copyright © 1964 Institute of Mathematical Statistics

Vol.35 • No. 3 • September, 1964
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