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

Abstract

Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $δ$-record if $X_n>\max\{X_1,...,X_{n-1}\}+δ$, 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 $δ\ne0$. In particular, taking $δ=-1$ we obtain a central limit theorem for the number of weak records.