Asymptotic normality for the counting process of weak records and δ-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 δ-record if X_n>max {X₁, …, X_n−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.