Present Value of a Renewal Process

Abstract

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.