On geometric record times

We analyze general time-homogeneous systems where the occurrences of events are Poissonian, and derive an integral equation for the Laplace transform of the geometric record times (proving existence and uniqueness for this equation). We then focus on Fréchet and Weibull systems which are governed by power-law Poissonian rate functions. For the geometric record times of these systems, we (i) compute the sequence of moments; and (ii) prove that the probability tails decay algebraically and compute the exponent governing the decay. This exponent turns out to be a non-linear function of the systems' power-law exponent.