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

Let $\{X_n,n\ge 1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $\delta$-record if $X_n>\max \{X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_{n-1}\}+\delta$, where $\delta$ is an integer constant. We use martingale arguments to show that the counting process of $\delta$-records among the first $n$ observations, suitably centered and scaled, is asymptotically normally distributed for $\delta \mathrm{\ne }0$. In particular, taking $\delta =-1$ we obtain a central limit theorem for the number of weak records.