论文标题
在最近的 - $ k $ - 离散随机变量的记录
On the recent-$k$-record of discrete random variables
论文作者
论文摘要
令$ x_1，〜x_2，\ cdots $为i.i.d随机变量的序列，应该按顺序观察到。序列中的$ n $ th值是$ k-record〜 value $，如果第一个$ n $值的$ k $（包括$ x_n $）至少与之一样大。令$ {\ bf r} _k $表示$ k $ - 记录值的有序集。著名的Ignatov定理指出，随机集$ {\ bf r} _k（k = 1,2，\ cdots）$独立于共同分布。在本文中，我们介绍了一张名为$ faster-k-record $（RKR）的新唱片：$ x_n $是$ j $ -rkr，如果恰好有$ j $ values的恰好最多与$ x_n $ in $ x_ {n-k}中的$ x_n $高，事实证明，RKR带来了许多有趣的问题以及一些新颖的属性，例如预测规则和泊松近似，这在本文中得到了证明。还提供了一个名为“ No Good Record”的应用程序通过LOV {Á} SZ本地引理。我们以与扫描统计数据有一些可能的联系来结束本文。
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{á}sz Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.
