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

Abstract

Let $X_1,~X_2,\cdots$ be a sequence of independent and identically distributed 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 $recent-k-record$ in this paper: $X_n$ is a $j$-recent-k-record 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 recent-k-record 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.