The paradox of the expected time until the next earthquake

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Abstract
We show analytically that the answer to the question, (can it be that) “The longer it has been since the last earthquake, the longer the expected time till the next?” depends crucially on the statistics of the fluctuations in the interval times between earthquakes. The periodic, uniform, semi-Gaussian, Rayleigh, and truncated statistical distributions of interval times, as well as the Weibull distributions with exponent greater than 1, all have decreasing expected time to the next earthquake with increasing time since the last one, for long times since the last earthquake; the lognormal and power-law distributions and the Weibull distributions with exponents smaller than 1 have increasing times to the next earthquake as the elapsed time since the last increases, for long elapsed times. There is an identifiable crossover between these models, which is gauged by the rate of fall-off of the long-term tail of the distribution in comparison with an exponential fall-off. The response to the question for short elapsed times is also evaluated. The lognormal and power-law distributions give one response for short elapsed times and the opposite for long elapsed times. Even the sampling of a finite number of intervals from a Poisson distribution will lead to an increasing estimate of time to the next earthquake for increasing elapsed time since the last one.