论文标题
与记录破坏相关的几何分布（1/2）
A Geometric(1/2) Distribution Associated with Record Breaking
论文作者
论文摘要
令$ x_i，i = 0,1，\ ldots $是一系列IID随机变量，它们的分布是连续的。与此序列关联的是序列$（i，x_i），i = 0,1，\ ldots $。令$ \ {（i，x_i）的帕累托的最佳元素表示$ {\ cal r} _ {n} $表示：i = 0，\ ldots，n \}。$我们指的是$ {\ cal r} _ {n} $的元素，如当前的记录，是时间$ n，$ n，$ n，$ n，我们define $ n = i define $ r_ r} _n \ vert，$此类记录的数量。观察$ r_n $具有$ \ {1，\ ldots，n+1 \} $作为其支持。当实现$（n，x_n）$时，它是$ \ {（i，x_i）〜：〜i = 0，\ ldots，n \} $和$ {\ cal r} _ { $ {\ cal b} _n = {\ cal r} _ {n-1} \ backslash {\ cal r} _ {n} $作为时间$n。$n。$n。$n。$n。$n。$n。$n。$n。$n。$n。$ n。 \ mbox {for} k = 0,1,2，\ ldots。$$
Let $X_i,i=0,1,\ldots$ be a sequence of iid random variables whose distribution is continuous. Associated with this sequence is the sequence $(i,X_i),i=0,1,\ldots$. Let ${\cal R}_{n}$ denote the set of Pareto optimal elements of $\{ (i,X_i):i=0,\ldots,n\}.$ We refer to the elements of ${\cal R}_{n}$ as the current records at time $n,$ and we define $R_n=\vert {\cal R}_n\vert,$ the number of such records. Observe that $R_n$ has $\{1,\ldots,n+1\}$ as its support. When $(n,X_n)$ is realized, it is a Pareto optimal element of $\{ (i,X_i)~:~i=0,\ldots,n\}$ and ${\cal R}_{n} \backslash (n,X_n) \subset {\cal R}_{n-1}.$ Then we refer to those elements of ${\cal B}_n = {\cal R}_{n-1} \backslash {\cal R}_{n}$ as the records broken at time $n.$ Let $B_n= \vert {\cal B}_n \vert.$ We show that $$P[B_n = k] \rightarrow 1/2^{k+1} \mbox{ for } k=0,1,2,\ldots.$$
