Sylvester's problem for random walks and bridges

TL;DR

This paper solves Sylvester's problem for certain random walks and bridges in Euclidean space, determining the probability that initial steps are in convex position, revealing a precise probability formula.

Contribution

It provides a novel exact probability formula for convex position of initial steps in random walks and bridges with specific distribution properties.

Findings

Probability that first d+2 steps are in convex position is 1 - 2/(d+1)!

Result holds for random walks with increments avoiding hyperplanes

Applicable to exchangeable random bridges of length d+2

Abstract

Consider a random walk in $\mathbb{R}^d$ that starts at the origin and whose increment distribution assigns zero probability to any affine hyperplane. We solve Sylvester's problem for these random walks by showing that the probability that the first $d+2$ steps of the walk are in convex position is equal to $1-\frac{2}{(d+1)!}$. The analogous result also holds for random bridges of length $d+2$, so long as the joint increment distribution is exchangeable.