论文标题
多维赌徒问题中的谐波度量
Harmonic measure in a multidimensional gambler's problem
论文作者
论文摘要
我们考虑在截短的锥体$ k_n $中随机步行，这是通过将超平面切成$ n $ $ n $将锥体$ k $切成薄片获得的。我们研究了该截短的锥体中绿色功能的行为，因为$ n $增加。使用这些结果，我们还获得了谐波度量的渐近行为。 获得的结果应用于Diaconis和Ethier（2022）研究的多维赌徒问题。特别是我们确认他们的猜想是，以特定顺序消除参与者的概率与布朗运动近似相同的渐近行为。我们还为这种近似值提供了这种概率的收敛速度。
We consider a random walk in a truncated cone $K_N$, which is obtained by slicing cone $K$ by a hyperplane at a growing level of order $N$. We study the behaviour of the Green function in this truncated cone as $N$ increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure. The obtained results are applied to a multidimensional gambler's problem studied by Diaconis and Ethier (2022). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.
