Expected mixed volumes of convex hulls of random walks and Lévy processes
Abstract
Let C1,…,Ck be the convex hulls of independent partial-sum processes in ℝd whose increments are exchangeable within each process. We express the expected mixed volume 𝔼 Vd(C1[m1],…,Ck[mk]), m1+⋯+mk=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.
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BibTeX
@misc{bolotin2026expected,
title = {{Expected mixed volumes of convex hulls of random walks and Lévy processes}},
author = {Artyom Bolotin and Dmitry Zaporozhets},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10228},
eprint = {2610.10228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10228},
}
RIS
TY - UNPB
TI - Expected mixed volumes of convex hulls of random walks and Lévy processes
AU - Artyom Bolotin
AU - Dmitry Zaporozhets
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10228
UR - https://arxiv.org/abs/2610.10228
AB - Let C_1,…,C_k be the convex hulls of independent partial-sum processes in ℝ^d whose increments are exchangeable within each process. We express the expected mixed volume 𝔼 V_d(C_1[m_1],…,C_k[m_k]), m_1+⋯+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.
ER -