Asymptotic normality for the counting process of weak records and delta-records in discrete models

Abstract

Let {X-n, n >= 1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call X-n a delta-record if X-n > max{X-1,..., Xn-1} + delta, where delta is an integer constant. We use martingale arguments to show that the counting process of 8-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for delta not equal 0. In particular, taking delta = -1 we obtain a central limit theorem for the number of weak records.