任意凸集中随机多边形的中心极限定理
Central limit theorems for random polygons in an arbitrary convex set
摘要：
研究了凸集$K\subset\mathbb{R}^2$中随机多边形的面积和顶点数的概率分布。我们的方法的新颖之处在于，它产生统一的估计所有凸集$K\subset\mathbb{R}^2$，而不施加任何正则性条件的边界$\partial K$。我们的主要结果是面积和顶点数的中心极限定理，解决了该领域的一个著名猜想。我们还得到了这两个泛函的期望和方差的增长有关的渐近结果。
英文摘要：
We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets $K\subset\mathbb{R}^2$ without imposing any regularity conditions on the boundary $\partial K$. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
