On the point process of near-record values

Let (Xn) be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution F . An observation Xn is a near-record if Xn ∈ (Mn−1 − a, Mn−1], where Mn = max{X1, . . . , Xn} and a > 0 is a parameter. We analyze the point process η on [0,∞) of near-record values from (Xn), showing that it is a Poisson cluster process. We derive the probability generating functional of η and formulas for the expectation, variance and covariance of the counting variables η(A), A ⊂ [0,∞). We also obtain strong convergence and asymptotic normality of η(t) := η([0, t]), as t → ∞, under mild tail-regularity conditions on F . For heavy-tailed distributions, with square-integrable hazard function, we show that η(t) grows to a finite random limit η(∞) and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data.