AI 中文总结
本文提出跨壁方法，重新证明随机游走凸包吸收概率的无分布性，推导其新概率表示与递推关系，还重新证明Wendel定理及一维Sparre Andersen定理。
AI 中文摘要
我们考虑d维随机游走的前n个部分和构成的凸包包含原点的概率。在增量满足对称可交换性及一般位置假设的条件下，该吸收概率是无分布的，且存在显式公式，此前Kabluchko、Vysotsky和Zaporozhets[Geom. Funct. Anal. 27 (2017)]利用超平面排列的特征多项式得到了该公式。本文给出一种不同的证明，基于我们在此处提出的跨壁方法。从增量的确定性构型出发，我们计数对应的部分和凸包包含原点的符号置换，证明该计数在增量的一般形变下保持不变，因此对自然零测例外集之外的所有构型均相同。在单个精心选择的构型处计算该不变量，将剩余计算简化为置换记录的枚举结合Wendel定理。我们的方法还重新证明了Wendel关于具有符号翻转不变联合分布的随机点凸包的定理，以及一维情况下的Sparre Andersen定理。最后，我们推导了随机游走凸包及其随机桥类似物的吸收概率的新概率表示和递推关系。
英文摘要
We consider the probability that the convex hull of the first $n$ partial sums of a $d$-dimensional random walk contains the origin. Under symmetric exchangeability of the increments and a general-position assumption, this absorption probability is distribution-free and admits an explicit formula, previously obtained by Kabluchko, Vysotsky and Zaporozhets [Geom. Funct. Anal. 27 (2017)] using characteristic polynomials of hyperplane arrangements. We give a different proof, based on a wall-crossing method which we develop here. Starting from a deterministic configuration of increments, we count the signed permutations for which the convex hull of the corresponding partial sums contains the origin and show that this count remains unchanged under generic deformations of the increments, and hence is the same for all configurations outside a natural exceptional set of measure zero. Evaluating the invariant at a single well-chosen configuration reduces the remaining calculation to the enumeration of permutation records combined with Wendel's theorem. Our method also reproves Wendel's theorem on convex hulls of random points with a sign-flip-invariant joint distribution and, in dimension one, Sparre Andersen's theorem. Finally, we derive new probabilistic representations and recurrence relations for the absorption probabilities of random-walk convex hulls and their random-bridge analogues.
Comments38 pages, no figures
