Convex hulls of random walks: conic intrinsic volumes approach

Sparre Andersen discovered a celebrated distribution-free formula for the probability of a random walk remaining positive up to a moment $n$. Kabluchko et al. expanded on this result by calculating the absorption probability for the convex hull of multidimensional random walks. They approached this by transforming the problem into a geometric one, which they then solved using Zaslavsky's theorem. We propose a completely different approach that allows us to directly derive the generating function for the absorption probability. The cornerstone of our method is the Gauss–Bonnet formula for polyhedral cones.