Exceedances of records

Given a sequence of random variables (rv’s) and a real function ψ , the ψ -exceedances form a subsequence consisting of those rv’s larger than a function, ψ , of the previous element of the subsequence. We present the basic distribution theory of ψ -exceedances for a sequence of independent and identically distributed rv’s. We give several examples and we study with more detail the case of exponential parents with ψ a linear function. The particular case of arithmetic exceedances is useful to describe the behavior of a type I counter when the arrival process of particles follows a non-homogeneous Poisson process. We also mention applications to destructive testing, early alert systems and the departure process of a M t / D / 1 / 1 queue.