Ratio limit theorem for renewal processes

TL;DR

This paper investigates the asymptotic behavior of survival probabilities in renewal processes modeling shock-induced failures, establishing a ratio limit theorem that links these probabilities to the initial renewal epoch under mild conditions.

Contribution

It introduces a new ratio limit theorem for renewal processes, connecting survival probabilities to the probability of the initial renewal epoch, with convergence rates depending on distribution regularity.

Findings

Ratio of survival probabilities converges to the probability of initial renewal epoch.

Convergence holds under mild regularity conditions on shock distributions.

Provides explicit rates of convergence based on distribution support and regularity.

Abstract

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precisions on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the pre-existing results on the behavior of summands of i.i.d. real random variables.