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

Abstract

Consider a sequence of partial sums $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(\xi_1,\dots,\xi_n)$ is exchangeable and a certain general position condition holds, we prove that the expected number of $k$-dimensional faces of $C_n$ is given by the formula $$ \mathbb E [f_k(C_n)] = \frac{2\cdot k!}{n!} \sum_{l=0}^{\infty}\genfrac{[}{]}{0pt}{}{n+1}{d-2l} \genfrac{\{}{\}}{0pt}{}{d-2l}{k+1}, $$ for all $0\leq k \leq d-1$, where $\genfrac{[}{]}{0pt}{}{n}{m}$ and $\genfrac{\{}{\}}{0pt}{}{n}{m}$ are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices $0\leq i_1<\dots <i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. 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 $\xi_k$'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 $A_{n-1}$ and $B_n$. 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.