Convex hulls of multidimensional random walks

Abstract

Let $S_k$ be a random walk in $\mathbb R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $\operatorname {conv} (S_1, \dots , S_n)$ 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: \begin{equation*} \mathbb E V_1 (\operatorname {conv} (0, S_1, \dots , S_n)) = \sum _{k=1}^n \frac {\mathbb E \|S_k\|}{k}, \end{equation*} 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.