TY - JOUR

T1 - Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

AU - Kabluchko, Zakhar

AU - Vysotsky, Vladislav

AU - Zaporozhets, Dmitry

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type An-1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1× ⋯ × B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as n→ ∞, in both cases of fixed and increasing dimension d.

AB - We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type An-1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1× ⋯ × B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as n→ ∞, in both cases of fixed and increasing dimension d.

UR - http://www.scopus.com/inward/record.url?scp=85021837269&partnerID=8YFLogxK

U2 - 10.1007/s00039-017-0415-x

DO - 10.1007/s00039-017-0415-x

M3 - Article

AN - SCOPUS:85021837269

VL - 27

SP - 880

EP - 918

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -