玩飞镖时该站在哪里?

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
Björn Franzén, J. Steif, Johan Wastlund
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引用次数: 1

摘要

本文分析了玩飞镖时应该站在哪里的问题。如果一个人站在距离$d>0$的地方,瞄准$a\in \mathbb{R}^n$,那么飞镖(由$\mathbb{R}^n$中的随机向量$X$建模)会击中$a+dX$给出的随机点。接下来,给定一个收益函数$f$,考虑$$ \sup_a Ef(a+dX) $$并问它是否在$d$中递减;也就是说,站得离目标近一点是否比站得离目标远一点好。也许令人惊讶的是,情况并非总是如此,了解这种情况何时发生或不发生是本文的目的。我们证明,如果$X$有一个所谓的自分解分布,那么对于任何收益函数来说,站得更近总是更好的。这个类包括所有的稳定发行版以及更多的发行版。另一方面,如果收益函数为$\cos(x)$,那么当且仅当特征函数$|\phi_X(t)|$在$[0,\infty)$上递减时,总是站得更近一些。然后我们将证明,如果至少有两个质点,那么使用$\cos(x)$站得更近并不总是更好。如果存在一个单点质量,我们可以找到一个不同的收益函数来获得这种现象。另一大类飞镖$X$,对于有界连续收益函数,它并不总是站得更近,这是具有紧凑支持的分布。这可以通过这样一个事实得到即这些分布的傅里叶变换在复平面上为零。这个论证在傅里叶变换为复零的情况下都成立。最后,我们分析了在卷积和/或极限条件下,越近越好这一性质是否封闭。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Where to stand when playing darts?
This paper analyzes the question of where one should stand when playing darts. If one stands at distance $d>0$ and aims at $a\in \mathbb{R}^n$, then the dart (modelled by a random vector $X$ in $\mathbb{R}^n$) hits a random point given by $a+dX$. Next, given a payoff function $f$, one considers $$ \sup_a Ef(a+dX) $$ and asks if this is decreasing in $d$; i.e., whether it is better to stand closer rather than farther from the target. Perhaps surprisingly, this is not always the case and understanding when this does or does not occur is the purpose of this paper. We show that if $X$ has a so-called selfdecomposable distribution, then it is always better to stand closer for any payoff function. This class includes all stable distributions as well as many more. On the other hand, if the payoff function is $\cos(x)$, then it is always better to stand closer if and only if the characteristic function $|\phi_X(t)|$ is decreasing on $[0,\infty)$. We will then show that if there are at least two point masses, then it is not always better to stand closer using $\cos(x)$. If there is a single point mass, one can find a different payoff function to obtain this phenomenon. Another large class of darts $X$ for which there are bounded continuous payoff functions for which it is not always better to stand closer are distributions with compact support. This will be obtained by using the fact that the Fourier transform of such distributions has a zero in the complex plane. This argument will work whenever there is a complex zero of the Fourier transform. Finally, we analyze if the property of it being better to stand closer is closed under convolution and/or limits.
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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