Abstract:

Consider the probability that the convex hull of a random walk in R^d does not absorb the origin by the time n. In dimension one this is simply the probability that there is no sign change, which is know to be equal to 2 (2n-1)!!/(2n)!! for any random walk with a symmetric density of increments. We prove a multidimensional distribution-free explicit tractable formula for the probability of absorption of the origin. Our idea is to show that the absorption problem is equivalent to a geometric problem on counting the number of Weyl chambers in R^n intersected by a generic linear subspace of codimension d. This method also applies to convex hulls of random walk bridges, and to the joint convex hulls of several symmetric random walks. In particular, we recover the Wendel formula for the absorption probability of the convex hull of i.i.d. random vectors in R^d with a symmetric density. If time allows, I will also discuss the asymptotics of the absorption probability in a fixed dimension and in the high-dimensional setting. This is a joint work with Zakhar Kabluchko (Munster) and Dmitry Zaporozhets (St. Petersburg).