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

Kabluchko, Zakhar; Vysotsky, Vladislav; Zaporozhets, Dmitry

doi:10.1016/j.aim.2017.09.002

Cited by 25 publications

(33 citation statements)

References 33 publications

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“…, d}. The following explicit formula for the expected face numbers of C n,d has been obtained in [42] relying on the methods of [43]:…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

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Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Kabluchko¹,

Marynych²

2021

Preprint

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Let ξ 1 , ξ 2 , . . . be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i := ξ 1 + • • • + ξ i , and let C n,d := conv(0, S 1 , S 2 , . . . , S n ) be the convex hull of the first n + 1 points it has visited. The polytope C n,d is called k-neighborly if for every indices 0We study the probability that C n,d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to ∞. There is an explicit formula for the expected number of k-dimensional faces of C n,d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C n,d . This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.

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“…, d}. The following explicit formula for the expected face numbers of C n,d has been obtained in [42] relying on the methods of [43]:…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…, S i d are linearly dependent is 0. For example, it is known from [43,Proposition 2.5] and [42,Example 1.1] that Conditions (Ex) and (GP) are satisfied if ξ 1 , . .…”

Section: It Follows Thatmentioning

confidence: 99%

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Kabluchko¹,

Marynych²

2021

Preprint

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“…The multidimensional case is much less studied, but the combinatorial lemma of [BNB63], from which one obtains the expected characteristics of the convex hull of (2D) random walks (like perimeter length or area) has been extended in various directions (and dimensions!) including [RFW17,KVZ17a,KVZ17b,VZ18]. Still in the realm of fluctuation theory, [Ber93] constructs (one-dimensional) random walks conditioned to stay positive through a bijection on permutations; this result is used here to study continuity of an EI process when it reaches its minimum.…”

Section: Definitionmentioning

confidence: 99%

Dini derivatives and regularity for exchangeable increment processes

Hernández

Bravo

2020

Trans. Amer. Math. Soc. Ser. B

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Let X be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that for any fixed t almost surely, lim sup hÑ0˘p X t`h´Xt q {h " 8 and lim sup hÑ0˘p X t`h´Xt q {h "´8. This extends a celebrated result of Rogozin for Lévy processes obtained in 1968 and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines p´8, 0q and p0, 8q are visited immediately for infinite variation EI processes (called upward and downward regularity). We also generalize the zero-one law of Millar for Lévy processes by showing continuity of X when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart, and Miller on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion considered in [DIM77] and broadened to a subclass of Lévy processes and EI processes by Chaumont and the second author. We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant known for Lévy processes found in [Ann. Prob. 40 (2012), pp. 1636-1674] and extend it to non-piecewise linear EI processes. Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for Lévy processes.

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“…3.4. The expected numbers of j-faces of the convex hull of a single Gaussian random walk or a Gaussian bridge in R d (even in a more general non-Gaussian setting) were already evaluated in [19]. Our general result on the angle sums of products of Schläfli orthoschemes yields a formula for the expected number of j-faces of the Minkowski sum of several convex hulls of Gaussian random walks or Gaussian random bridges.…”

Section: Introductionmentioning

confidence: 95%

Angle Sums of Schläfli Orthoschemes

Godland

Kabluchko

2021

Discrete Comput Geom

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We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

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

Cited by 25 publications

References 33 publications

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Dini derivatives and regularity for exchangeable increment processes

Angle Sums of Schläfli Orthoschemes

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