On the point process of near-record values

Abstract

Let be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution . An observation is a near-record if , where and is a parameter. We analyze the point process on of near-record values from , showing that it is a Poisson cluster process. We derive the probability generating functional of and formulas for the expectation, variance and covariance of the counting variables . We also obtain strong convergence and asymptotic normality of , as , under mild tail-regularity conditions on . For heavy-tailed distributions, with square-integrable hazard function, we show that grows to a finite random limit and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data.