Near-Record Values in Discrete Random Sequences

Given a sequence (Xn) of random variables, Xn is said to be near-record if Xn∈(Mn−1−a,Mn−1], where Mn=max{X1, …, Xn} and a>0 parameter. We investigate the point process η on [0,∞) values from an integer-valued, independent identically distributed sequence, showing that it Bernoulli cluster process. derive probability generating functional formulas for expectation, variance covariance counting variables η(A),A⊂[0,∞). also strong convergence asymptotic normality η([0,n]), as n→∞, under mild regularity conditions distribution observations. For heavy-tailed distributions, with square-summable hazard rates, we prove η([0,n]) grows finite limit compute its function. present examples application our results particular covering wide range behaviours in terms their right tails.