  We study the asymptotic distribution for the occurrence time of the next
large earthquake, by knowing the last large seismic event occurred a long time
ago. We prove that, under reasonable conditions, such a distribution is
asymptotically exponential with a rate depending on the asymptotic slope of the
cumulative intensity function corresponding to a non-homogeneous Poisson
process. Moreover, as it is not possible to obtain an empirical cumulative
distribution function for the waiting time of the next large earthquake, a
random cumulative function based on existing data is stated. We demonstrate
that analogous results to the theorems of Glivenko-Cantelli and Kolmogorov are
satisfied by this random cumulative function. We conduct a simulation study for
detecting in what scenario the approximate distribution of the studied elapsed
time performs well. Finally, a real-world data analysis is carried out to
illustrate the potential applications of our proposal.
