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

Kabluchko, Zakhar; Vysotsky, Vladislav; Zaporozhets, Dmitry

Zusammenfassung

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)]=[Formula presented]∑l=0∞[[Formula presented]]{[Formula presented]}, for all 0≤k≤d−1, where [[Formula presented]] and {[Formula presented]} are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0≤i1<…