论文标题
玩飞镖时在哪里站立？
Where to stand when playing darts?
论文作者
论文摘要
本文分析了一个人在玩飞镖时应立即站立的问题。如果一个人站在距离$ d> 0 $上，而目标却以$ a \ in \ mathbb {r}^n $为目标，则该飞镖（由$ \ mathbb {r}^n $在$ \ mathbb {r}^n $中建模）命中了由$ a+dx $给出的随机点。接下来，给定收益功能$ f $，一个人考虑$$ \ sup_a ef（a+dx）$$，并询问这是否在$ d $中减少；即，是否更好地站在距离目标方面是更好的。也许令人惊讶的是，本文的目的并非总是如此，并且了解何时或不发生的情况是。 我们表明，如果$ x $具有所谓的自我补充分布，那么最好靠近任何回报功能。该课程包括所有稳定的分布以及更多分布。 另一方面，如果收益函数为$ \ cos（x）$，那么只有当特征函数$ | ϕ_x（t）| $在$ [0，\ infty）$上减少时，始终更好地站起来。然后，我们将证明，如果至少有两个点质量，那么使用$ \ cos（x）$站立并不总是更好。如果有一个点质量，可以找到不同的回报函数来获得这种现象。 另一个大型的飞镖$ x $，其中有一个有限的连续回报功能，而靠近的持续收益功能并不总是更好的是具有紧凑的支持的发行版。这将通过使用以下事实，即这种分布的傅立叶变换在复合面中为零。只要有傅立叶变换的复杂零，该参数就会起作用。 最后，我们分析在卷积和/或限制下，靠近的属性是否更好地关闭。
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 $|ϕ_X(t)|$ 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.
