Expected mixed volumes of convex hulls of random walks and Lévy processes

Source abstract

Let $C_1,\ldots,C_k$ be the convex hulls of independent partial-sum processes in $\mathbb{R}^d$ whose increments are exchangeable within each process. We express the expected mixed volume $\mathbb{E} V_d(C_1[m_1],\ldots,C_k[m_k])$, $m_1+\cdots+m_k=d$, through mean absolute determinants of disjoint block sums of the increments; no general-position assumption is needed. For random walks with i.i.d. integrable increments the block sums are independent, and the formula extends the expected-volume formula of Barndorff-Nielsen and Baxter and of Vysotsky and Zaporozhets to mixed volumes. We then prove a continuous-time counterpart: for independent Lévy processes with finite first moments, the expected mixed volume of the closed convex hulls of their paths is an explicit integral of mean absolute determinants over a product of simplices. For symmetric stable processes the integral is evaluated in terms of the associated zonoids. As a geometric application, we compute the mean mixed volume of random projections of mutually orthogonal canonical orthoschemes.