Near-Record Values in Discrete Random Sequences

Given a sequence ( X n ) of random variables, X n is said to be a near-record if X n ∈ ( M n − 1 − a , M n − 1 ] , where M n = max { X 1 , … , X n } and a > 0 is a parameter. We investigate the point process η on [ 0 , ∞ ) of near-record values from an integer-valued, independent and identically distributed sequence, showing that it is a Bernoulli 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 derive the strong convergence and asymptotic normality of η ( [ 0 , n ] ) , as n → ∞ , under mild regularity conditions on the distribution of the observations. For heavy-tailed distributions, with square-summable hazard rates, we prove that η ( [ 0 , n ] ) grows to a finite random limit and compute its probability generating function. We present examples of the application of our results to particular distributions, covering a wide range of behaviours in terms of their right tails.