Convex hulls of random walks: expected number of faces and face probabilities

Consider a sequence of partial sums Si=?1+…+?i, 1=i=n, starting at S0=0, whose increments ?1,…,?n 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 (?1,…,?n) 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=08[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<…