On geometric record times

We study geometric record times in continuous-time systems where events of random (positive) magnitudes occur stochastically. Namely, given that the current record level is x, and given a parameter k>1, we address the following question: how long would we have to wait till the occurrence of a record event whose magnitude is at least k-times greater than the magnitudes of all the record events preceding it? 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.