Convex hulls of multidimensional random walks
Convex hulls of multidimensional random walks
复制标题
多维随机游走的凸包
DOI：
10.1090/tran/7253
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发表时间：
2015
影响因子：
1.3
通讯作者：
D. Zaporozhets
中科院分区：
文献类型：
--
作者：
V. Vysotsky;D. Zaporozhets
设\(S_k\)是\(\mathbb{R}^d\)中的一个随机游走，使得其增量分布不给超平面分配质量。我们研究游走的前\(n\)步的凸包\(\text{conv}(S_1,\ldots,S_n)\)不包含原点的概率\(p_n\)。通过提供一个显式公式，我们表明对于平面上对称分布的随机游走，\(p_n\)不依赖于增量分布。这扩展了斯帕雷·安德森（1949年）的著名结果，即满足上述连续性和对称性假设的一维随机游走以一个与分布无关的概率保持为正。我们还找到了对于任何具有零均值平方可积增量的平面随机游走，当\(n\to\infty\)时\(p_n\)的渐近性。
我们进一步从平面情形发展我们的方法，以研究任意维度\(d\geq2\)中随机游走凸包的一大类几何特征。特别地，我们给出了面的数量的期望值、体积、表面积以及其他内蕴体积的公式，包括以下关于平面游走周长的斯皮策 - 威多姆公式（1961年）的多维推广：
\(E V_1(\text{conv}(0,S_1,\dots,S_n))=\sum_{k = 1}^n\frac{E\|S_k\|}{k}\)，
其中\(V_1\)表示第一个内蕴体积，它与平均宽度成正比。
这些结果在几何中有应用，特别地，暗示了高和维塔莱（2001年）对于特殊路径 - 单形（称为规范正割体，它是维纳螺旋的闭凸包的有限维近似）的内蕴体积的公式。此外，这些单形的球面内蕴体积与概率\(p_n\)之间有直接联系。
我们还对随机游走桥的凸包以及更一般地对可交换随机向量的部分和证明了类似的结果。
Let $S_k$ be a random walk in $R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, $p_n$ does not depend on the distribution of increments. This extends the well known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of $p_n$ as $n \to \infty$ for any planar random walk with zero mean square-integrable increments.
We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension $d \ge 2$. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer--Widom formula (1961) on the perimeter of planar walks: $$ E V_1 (conv(0, S_1, \dots, S_n)) = \sum_{k=1}^n \frac{E \|S_k\|}{k}, $$ where $V_1$ denotes the first intrinsic volume, which is proportional to the mean width.
These results have applications to geometry, and in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities $p_n$.
We also prove similar results for convex hulls of random walk bridges, and more generally, for partial sums of exchangeable random vectors.
随机游走的凸包、超平面排列和 Weyl 室
影响因子：
2.2
作者：
Zakhar Kabluchko;Vladislav Vysotsky;Dmitry Zaporozhets
通讯作者：
Dmitry Zaporozhets