On the recent$k$record of discrete random variables
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 $krecord~value$ 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 $recentkrecord$ (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 $X_{nk},~X_{nk+1},\cdots,~X_{n1}$. 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{á}sz Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.06791
 Bibcode:
 2022arXiv220806791L
 Keywords:

 Mathematics  Probability;
 62H10;
 62B10