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

TL;DR

This paper proves that the counting process of elta-records in i.i.d. discrete models follows an asymptotic normal distribution, extending to weak records when elta=-1, using martingale techniques.

Contribution

It introduces a martingale-based approach to establish asymptotic normality for elta-records in discrete i.i.d. models, including weak records.

Findings

Asymptotic normality of elta-record counting process for eltane 0

Central limit theorem for weak records (elta=-1)

Martingale arguments effectively prove distributional convergence

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 $\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, 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 $\delta\ne0$. In particular, taking $\delta=-1$ we obtain a central limit theorem for the number of weak records.