Abstract

Let {Xn, n ≥ 1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call Xn a δ-record if Xn > max(X1,..., Xn-1) + δ where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ ≠ 0. In particular, taking δ = -1 we obtain a central limit theorem for the number of weak records. © 2007 ISI/BS.