TY - JOUR

T1 - Expected volumes of gaussian polytopes, external angles, and multiple order statistics

AU - Kabluchko, Zakhar

AU - Zaporozhets, Dmitry

PY - 2019/8/1

Y1 - 2019/8/1

N2 - Let X1,…,Xn be a standard normal sample in Rd . We compute exactly the expected volume of the Gaussian polytope conv [X1,…,Xn], the symmetric Gaussian polytope conv [±X1,…,±Xn], and the Gaussian zonotope [0,X1 ]+···+[0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1,…,lnXn] and conv [±l1X1,…,±lnXn], where l1,…,ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1,…,ξn given that the maximum has multiplicity k. Namely, we show that (Formula Presented) where ξ(1) ≤···≤ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1,…,ξn with their absolute values.

AB - Let X1,…,Xn be a standard normal sample in Rd . We compute exactly the expected volume of the Gaussian polytope conv [X1,…,Xn], the symmetric Gaussian polytope conv [±X1,…,±Xn], and the Gaussian zonotope [0,X1 ]+···+[0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1,…,lnXn] and conv [±l1X1,…,±lnXn], where l1,…,ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1,…,ξn given that the maximum has multiplicity k. Namely, we show that (Formula Presented) where ξ(1) ≤···≤ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1,…,ξn with their absolute values.

KW - Asymptotics

KW - Burgers festoon

KW - Expected volume

KW - External angles

KW - Extreme-value theory

KW - Gaussian polytope

KW - Intrinsic volumes

KW - Order statistics

KW - Regular cross-polytope

KW - Regular simplex

KW - Symmetric Gaussian polytope

UR - http://www.scopus.com/inward/record.url?scp=85071159561&partnerID=8YFLogxK

U2 - 10.1090/tran/7708

DO - 10.1090/tran/7708

M3 - Article

AN - SCOPUS:85071159561

VL - 372

SP - 1709

EP - 1733

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -