Convex hulls of Lévy processes

Molchanov, Ilya; Wespi, Florian

doi:10.1214/16-ecp19

Cited by 14 publications

(24 citation statements)

References 3 publications

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“…Let U be a random vector distributed uniformly over the unit sphere S d−1 . Then for any sequence h n tending to infinity such that h n = o(n), we have Similarly to (19), this gives (see Section 6) the asymptotics…”

Section: The Main Tool For Multidimensional Random Walksmentioning

confidence: 83%

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Convex hulls of multidimensional random walks

Vysotsky

Zaporozhets

2018

Trans. Amer. Math. Soc.

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Abstract. Let S k be a random walk in R d such that its distribution of increments does not assign mass to hyperplanes. We study the probability p n that the convex hull conv(S 1 , . . . , S n ) of the first n steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, p n does not depend on the distribution of increments. This extends the well known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of p n as n → ∞ for any planar random walk with zero mean square-integrable increments.We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension d ≥ 2. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer-Widom formula (1961) on the perimeter of planar walks:where V 1 denotes the first intrinsic volume, which is proportional to the mean width. These results have applications to geometry, and in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities p n .We also prove similar results for convex hulls of random walk bridges, and more generally, for partial sums of exchangeable random vectors.

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Section: The Main Tool For Multidimensional Random Walksmentioning

confidence: 83%

“…Consequently, the indices with n − i d ≤ c n do not contribute to the asymptotics in (19) and (21). On the other hand,…”

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confidence: 96%

Convex hulls of multidimensional random walks

Vysotsky

Zaporozhets

2018

Trans. Amer. Math. Soc.

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“…Convex hulls of random walks and their applications to Brownian motions and Lévy processes were much studied; see, e.g., [2,3,5,14,15,16,17,22,26,29,30]. These papers concentrate mostly on functionals like the volume and the perimeter, which are not distribution-free.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…Recall that g : R n → R n is a linear operator given by (17), (18), (19). The columns of the matrix Ag are…”

Section: Absorption Probability and Subspaces Intersecting Weyl Chambmentioning

confidence: 99%

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

Kabluchko

Vysotsky

Zaporozhets

2017

Advances in Mathematics

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Consider a sequence of partial sums S i = ξ 1 + · · · + ξ i , 1 ≤ i ≤ n, starting at S 0 = 0, whose increments ξ 1 , . . . , ξ n are random vectors in R d , d ≤ n. We are interested in the properties of the convex hull C n := Conv(S 0 , S 1 , . . . , S n ). Assuming that the tuple (ξ 1 , . . . , ξ 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 formulawhere n m and n m are Stirling numbers of the first and second kind, respectively.Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < · · · < i k+1 ≤ n, the points S i1 , . . . , S i k+1 form a k-dimensional face of Conv(S 0 , S 1 , . . . , 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 ξ 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.2010 Mathematics Subject Classification. Primary: 52A22, 60D05, 60G50; secondary: 60G09, 52C35, 20F55, 52B11, 60G70.

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“…In two dimensions, exact expressions for the values of geometric quantities such as the mean perimeter or the mean area of the convex hull have been found for Brownian motion [14][15][16][17] as well as for other types of stochastic motions [18][19][20], for multi-walker systems [21] and for confined configurations [22,23]. In some cases, equivalent results exist in higher dimensional space [19,[24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Facets on the convex hull ofd-dimensional Brownian and Lévy motion

Randon‐Furling

Wespi

2017

Phys. Rev. E

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For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d ≥ 3, we establish an exact formula for the average number of (d − 1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d = 3, a case which is of particular interest for applications in biophysics, chemistry and polymer science.We also show that the asymptotical average number of facets behaves as, where T is the total duration of the motion and ∆t is the minimum time lapse separating points that define a facet.

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Convex hulls of Lévy processes

Cited by 14 publications

References 3 publications

Convex hulls of multidimensional random walks

Convex hulls of multidimensional random walks

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

Facets on the convex hull ofd-dimensional Brownian and Lévy motion

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