打破多变量记录
Breaking Multivariate Records
摘要：
对于一系列的i.i.d. $d$ -具有独立连续分布坐标的维随机向量，假设序列中的第$n$个观测值设置记录，如果它在每个坐标中不被先前的观测值支配;对于$j \leq n$，假设第$j$个观测值是时间$n$处的当前记录，如果它在每个坐标中不被任何第一个$n$观测值支配;并且如果第$n$观测设置记录并且存在在时间$n - 1$但不是在时间$n$是当前记录的$k$观测，则说第$n$观测中断$k$记录。对于一般的尺寸$d$，我们确定，证明，（帕累托）记录的数量的渐近条件分布的观察打破了观察设置一个记录。修复$d$，并使${\mathcal K}(d)$为具有此分布的随机变量。证明了${\mathcal K}(d)$的（右）尾满足\[ {\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - Ω\!\left( k^{(d - 1) / (d^2 + d - 3)} \right) \right]\ \ \mbox{as $k \to \infty$} \]和\[ {\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\ \ \mbox{as $k \to \infty$}. \]。当$d = 2$时，${\mathcal K}(2)$的泊松过程描述与Fill [Comb. Probab. Comput. 30（2021）105- 123]，${\mathcal K}(2)$具有与${\mathcal G} - 1$相同的分布，其中${\mathcal G} \sim \mbox{Geometric$（1/2）$}$。注意，${\mathbb P}({\mathcal K}(d) \geq k)$的下限意味着对于任何$d \geq 3$，${\mathcal K}(d)$的分布都不是（移位的）几何分布。我们表明，${\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-Θ(d)]$作为$d \to \infty$，特别是，${\mathcal K}(d) \to 0$的概率为$d \to \infty$。
英文摘要：
For a sequence of i.i.d. $d$-dimensional random vectors with independent continuously distributed coordinates, say that the $n$th observation in the sequence sets a record if it is not dominated in every coordinate by an earlier observation; for $j \leq n$, say that the $j$th observation is a current record at time $n$ if it has not been dominated in every coordinate by any of the first $n$ observations; and say that the $n$th observation breaks $k$ records if it sets a record and there are $k$ observations that are current records at time $n - 1$ but not at time $n$. For general dimension $d$, we identify, with proof, the asymptotic conditional distribution of the number of (Pareto) records broken by an observation given that the observation sets a record. Fix $d$, and let ${\mathcal K}(d)$ be a random variable with this distribution. We show that the (right) tail of ${\mathcal K}(d)$ satisfies \[ {\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - Ω\!\left( k^{(d - 1) / (d^2 + d - 3)} \right) \right]\ \ \mbox{as $k \to \infty$} \] and \[ {\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\ \ \mbox{as $k \to \infty$}. \] When $d = 2$, the description of ${\mathcal K}(2)$ in terms of a Poisson process agrees with the main result from Fill [Comb. Probab. Comput. 30 (2021) 105--123] that ${\mathcal K}(2)$ has the same distribution as ${\mathcal G} - 1$, where ${\mathcal G} \sim \mbox{Geometric$(1/2)$}$. Note that the lower bound on ${\mathbb P}({\mathcal K}(d) \geq k)$ implies that the distribution of ${\mathcal K}(d)$ is NOT (shifted) Geometric for any $d \geq 3$. We show that ${\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-Θ(d)]$ as $d \to \infty$; in particular, ${\mathcal K}(d) \to 0$ in probability as $d \to \infty$.
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