# A stochastic methodology for risk assessment of a large earthquake when a long time has elapsed

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## Abstract

We propose a stochastic methodology for risk assessment of a large earthquake when a long time has elapsed from the last large seismic event. We state an approximate probability distribution for the occurrence time of the next large earthquake, by knowing that the last large seismic event occurred a long time ago. We prove that, under reasonable conditions, such a distribution is exponential with a rate depending on the asymptotic slope of the cumulative intensity function corresponding to a nonhomogeneous Poisson process. As it is not possible to obtain an empirical cumulative distribution function of the waiting time for the next large earthquake, an estimator of its cumulative distribution function based on existing data is derived. We conduct a simulation study for detecting scenario in which the proposed methodology would perform well. Finally, a real-world data analysis is carried out to illustrate its potential applications, including a homogeneity test for the times between earthquakes.

### Keywords

Earthquake data analysis Exponential and gamma distributions Maximum-likelihood method Monte Carlo simulation Nonhomogeneous Poisson process## Notes

### Acknowledgments

The authors thank the editors and referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. This research was partially supported by FONDECYT 1160868 Grant of CONICYT-Chile.

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