Consider a sequence of partial sums Si=ξ1+…+ξi, 1≤i≤n, starting at S0=0, whose increments ξ1,…,ξn are random vectors in Rd, d≤n. We are interested in the properties of the convex hull Cn:=Conv(S0,S1,…,Sn). Assuming that the tuple (ξ1,…,ξn) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of Cn is given by the formula
E[fk(Cn)]=2⋅k!n!∑l=0∞[n+1d−2l]{d−2lk+1},
for all 0≤k≤d−1, where [nm] and {nm} are Stirling numbers of the first and second kind, respectively.
Further, we compute explicitly the probability that for given indices 0≤i1<…<ik+1≤n, the points Si1,…,Sik+1 form a k-dimensional face of Conv(S0,S1,…,Sn). This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξk's.
The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types An−1 and Bn. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position
