Convex Hulls of Multidimensional Random Walks

Let Sk be a random walk in R such that its distribution of increments does not assign mass to hyperplanes. We study the probability pn that the convex hull conv(S1, . . . , Sn) 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, pn 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 pn as n→∞ 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 ≥ 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: EV1(conv(0, S1, . . . , Sn)) = n ∑