On the recent-$k$-record of discrete random variables

TL;DR

This paper introduces a new type of record called recent-$k$-record in i.i.d. sequences, explores its properties, and presents applications including prediction rules, Poisson approximation, and a 'No Good Record' problem using the Lovász Local Lemma.

Contribution

It proposes the recent-$k$-record concept, analyzes its properties, and develops novel results such as prediction rules and applications in probability and combinatorics.

Findings

Introduction of recent-$k$-record concept

Development of prediction rules for RkR

Application to 'No Good Record' problem

Abstract

Let $X_1,~X_2,\cdots$ be a sequence of i.i.d random variables which are supposed to be observed in sequence. The $n$th value in the sequence is a $k-record~value$ if exactly $k$ of the first $n$ values (including $X_n$) are at least as large as it. Let ${\bf R}_k$ denote the ordered set of $k$-record values. The famous Ignatov's Theorem states that the random sets ${\bf R}_k(k=1,2,\cdots)$ are independent with common distribution. We introduce one new record named $recent-k-record$ (RkR in short) in this paper: $X_n$ is a $j$-RkR if there are exactly $j$ values at least as large as $X_n$ in $X_{n-k},~X_{n-k+1},\cdots,~X_{n-1}$. It turns out that RkR brings many interesting problems and some novel properties such as prediction rule and Poisson approximation which are proved in this paper. One application named "No Good Record" via the Lov{\'a}sz Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.