On the convex hull of symmetric stable processes

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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“…For a standard Brownian motion, the intrinsic volumes V 1 and V 2 were found in the earlier paper by Kampf et al [15].…”

Section: Corollary 3 (Expected Intrinsic Volumes) We Havementioning

confidence: 56%

“…Arguing as above and using (14) instead of (15), one can easily show that (45) also holds for a random walk bridge of length n+1. For any sequence b n such that b n ∼ (log n)…”

mentioning

confidence: 98%

Convex hulls of multidimensional random walks

Vysotsky

Zaporozhets

2018

Trans. Amer. Math. Soc.

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“…Much is known about the properties of h t : see e.g. [2,4,6,13]. We also set ℓ t := L(h t ), and a t := A(h t ), the perimeter length and area of the standard Brownian convex hull.…”

Section: Brownian Convex Hulls As Scaling Limitsmentioning

confidence: 99%

“…From (13) ] is uniformly bounded. Hence n −2 A 2 n is uniformly integrable, and we deduce convergence of n −2 Var A n in Corollary 2.6.…”

Section: Variance Asymptoticsmentioning

confidence: 99%

Convex hulls of random walks and their scaling limits

Wade

2015

Stochastic Processes and their Applications

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For the perimeter length and the area of the convex hull of the first n steps of a planar random walk, we study n→∞ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero

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“…That is, the smallest convex set enclosing all points visited by the process. 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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On the convex hull of symmetric stable processes

Cited by 28 publications

References 13 publications

Convex hulls of multidimensional random walks

Convex hulls of multidimensional random walks

Convex hulls of random walks and their scaling limits

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

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