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 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 extensions for future work.

Keywords

• Ignatov’s theorem
• recent-k-record
• Poisson approximation
• the Lovász Local Lemma

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