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

We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^d}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A_{n-1}}$$\end{document} (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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B_1\times \cdots \times B_1}$$\end{document} 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→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \to \infty}$$\end{document}, in both cases of fixed and increasing dimension d.

P[S 1 > 0, . . . , S n > 0] = (2n − 1)!! 2 n n! = 1 2 2n 2n n . (2) We will refer to the condition that P[ξ 1 ∈ H] = 0 for every affine hyperplane H ⊂ R d as to the general position assumption since it implies that any d random vectors among S 1 , . . . , S n are linearly independent a.s.; see Proposition 2.5 below. We will impose this or similar assumptions in most of our results. However, even without it we can show that the absorption probability for any n-step symmetric random walk 1 (for which P[ξ 1 ∈ H] may be positive) is lower-bounded by the right-hand side in (1); see Proposition 2.12 below.

The equivalent geometric problems.
Our proof rests on a newly established direct connection between the probabilistic problem and an equivalent geometric problem concerning Weyl chambers. We solve the geometric problem using the theory of hyperplane arrangements and find the absorption probabilities explicitly. This method is entirely different from that of [VZ17], and it even requires a noticeable effort to check that the formulas for the absorption probabilities match for d = 2 (we will not include the details but wish to thank the anonymous referee for showing us this derivation).
It turns out that under the assumptions of Theorem 1.1, we have where N n,d is the constant number of Weyl chambers intersected by a generic nonrandom linear subspace of R n of codimension d. The exact meaning of "generic" will be explained in Sect. 3, where we compute the value of N n,d using the formulas of Whitney and Zaslavsky from the theory of hyperplane arrangements; see Theorem 3.4.

884
Z. KABLUCHKO ET AL. GAFA on R n by permuting the coordinates in arbitrary way and multiplying any number of coordinates by −1. This is a finite reflection group generated by reflections along the hyperfaces of any Weyl chamber of type B n . Every Weyl chamber W n of type B n is a fundamental region for the action of B n : this means that the sets gW n , g ∈ B n , are disjoint and their closures constitute the entire R n . We refer to [GB85] for an introduction to finite reflection groups. Our method actually applies to convex hulls of not only random walks with i.i.d. increments but also of any sequence of partial sums S k = ξ 1 + · · · + ξ k , k = 1, . . . , n, if their increments ξ 1 , . . . , ξ n are possibly dependent random vectors in R d whose joint distribution is invariant under the action of B n , that is (ξ 1 , . . . , ξ n ) d = (ε 1 ξ σ(1) , . . . , ε n ξ σ(n) ) ( 4 ) for any permutation σ(1), . . . , σ(n) of 1, . . . , n and any ε 1 , . . . , ε n ∈ {−1, 1}. In this case we say that the tuple (ξ 1 , . . . , ξ n ) is symmetrically exchangeable. There are many important examples of such tuples with non-i.i.d. entries: say, for d = 1, the tuple of coordinates of any rotationally invariant non-Gaussian random vector in R n is so. The exact statement of Theorem 1.1 extended to symmetrically exchangeable increments is given in Theorem 2.3 of the next section. It turns out that our approach can be generalized to solve the absorption problems for partial sums of increments whose joint distribution is invariant under the action of other finite groups. These are the reflection groups of types A n−1 and D n , and direct products of finite reflection groups. Let us explain.
The Coxeter group A n−1 is the symmetry group of the regular simplex (defined as the convex hull of the standard basis vectors in R n ). The n! elements of this group act on R n by permuting the coordinates. The tuple (ξ 1 , . . . , ξ n ) of random vectors in R d is called exchangeable if its distribution is invariant under the action of A n−1 , that is (ξ 1 , . . . , ξ n ) d = (ξ σ(1) , . . . , ξ σ(n) ) ( 5 ) for any permutation σ(1), . . . , σ(n) of 1, . . . , n. We will use the standard notation Sym(n) for the symmetric group on {1, . . . , n} of such permutations. The action of A n−1 leaves the hyperplane x 1 + · · · + x n = 0 invariant. Restricting the action of A n−1 to this hyperplane allows us to apply the method described above in Sect. 1.2 to the convex hulls Conv(S 1 , . . . , S n−1 ) of partial sums with exchangeable increments that satisfy the condition ξ 1 + · · · + ξ n = 0 a.s. This covers the absorption problem for random walk bridges under the corresponding general position assumption. The respective analogue of (3) rests on counting the number of Weyl chambers of type A n−1 intersected by a generic subspace of codimension d of the hyperplane x 1 + · · · + x n = 0. The exact statement is given in Theorem 2.1.
The Coxeter group D n is the subgroup of B n of index two whose action on R n changes signs of an even number of coordinates. The corresponding Theorem 2.7, concerning the joint distribution of increments invariant under the action of D n , GAFA CONVEX HULLS OF RANDOM WALKS AND WEYL CHAMBERS 885 applies to the convex hulls of symmetric random walks that are allowed to choose the sign of the last jump; see the discussion in Sect. 2. Thus, all the results discussed above in Sect. 1.2 are proved for the convex hulls of partial sums of increments whose joint distribution is invariant under the action of one of the finite reflection groups A n−1 , B n , or D n . The corresponding equivalent geometric statements concern Weyl chambers of the respective types.
Finally, we consider the case when the increments are invariant under the action of a direct product of reflection groups. We restrict ourselves to direct products of groups of type B. The corresponding Theorem 2.9 solves the absorption problem for the joint convex hull of several symmetric random walks with possibly different number of steps (under the corresponding general position assumption).
In the particular case when all the random walks have the same distribution of increments and each walk has only one step, Theorem 2.9 implies the well-known result of Wendel [Wen62]: if ξ 1 , . . . , ξ n are i.i.d. random vectors in R d with an absolutely continuous centrally symmetric distribution, then Thus, our approach brings together the classical distribution-free results (2) and (6) by Sparre Andersen and Wendel, respectively, on symmetrically distributed random variables.
Let us explain the structure of the paper. Section 2 contains the explicit formulas for the absorption probabilities under the different types of increments. These results are proved in Sect. 6. In Sects. 3 and 4 we provide some basic facts from the theory of hyperplane arrangements and conic convex geometry, and prove our new results on the geometric problems equivalent to the absorption problem. The asymptotic analysis of absorption probabilities is given in Sect. 5. We conclude the paper with the list of open questions.

Main results: convex hulls of random walks and bridges
In this section we present the explicit formulas for the absorption probabilities for the partial sums with the joint distribution of their increments ξ 1 , . . . , ξ n ∈ R d invariant under the action of the finite reflection groups A n−1 , B n , D n , and the analogous result for the invariance under direct products. 2.1 Type A n−1 : random walk bridges. The Coxeter group A n−1 is the symmetric group Sym(n), which acts on R n by permuting the coordinates. The number of elements in this group is n!. The action of this group leaves the hyperplane invariant. This explains why the subscript n−1 rather than n appears in the standard notation A n−1 . Note that the group A n−1 is the symmetry group of the regular simplex with n vertices, i.e. the convex hull of the standard basis in R n .
Theorem 2.1. Let (ξ 1 , . . . , ξ n ) be an exchangeable tuple (see (5)) of random vectors in R d with partial sums S 1 , . . . , S n . Assume that S n = 0 a.s., n ≥ d + 1, and any d random vectors among S 1 , . . . , S n−1 are linearly independent a.s. Then where n k are the Stirling numbers of the first kind defined by the formula with the convention that n k = 0 for k / ∈ {1, . . . , n}.
Remark 2.2. The sum in (7) is (as well as many other sums of a similar type appearing below) contains only finitely many non-zero terms. Combining (7) with the identity n 1 + n 3 + · · · = n 2 + n 4 + · · · = n! 2 (9) (which can be obtained by taking t = ±1 in (8)), yields the following formula for the probability of non-absorption: In the one-dimensional case d = 1 we obtain from (10) that the probability that S 1 , . . . , S n−1 do not change their sign is since n 1 = (n − 1)!. In fact, Sparre Andersen [And53, Corollary 2] showed that the probability of staying positive and the probability of staying negative are 1/n each. Theorem 2.1 can be viewed as a multidimensional generalization of this classical formula.
The number 1 n! n k turns out to be the kth conic intrinsic volume of the Weyl chamber of type A n−1 , as shown below in Sect. 4.3. It can be also interpreted as the where δ 1 , . . . , δ n are independent random variables (the record indicators) with δ i ∼ Bernoulli( 1 i ). It is remarkable that Theorem 2.1 (as well as the other similar theorems stated below) is distribution-free, that is the probability in (7) does not depend on the distribution of ξ 1 , . . . , ξ n . No moment conditions on the random vectors are imposed.
Let us stress that without the general position condition imposed in Theorem 2.1, the absorption probabilities become distribution-dependent. For example, for the bridge of a simple random walk S k on Z (which makes jumps ±1 with probability 1/2), it is known that for any even n, it holds which is clearly different from (11).
2.2 Type B n : symmetric random walks. The Coxeter group B n is the symmetry group of the regular cube [−1, 1] n (or of its dual, the regular crosspolytope). The elements of this group act on R n by permuting the coordinates in arbitrary way and multiplying any number of coordinates by −1. The number of elements of this group is 2 n n!.
It follows that the probability of non-absorption is given by The next proposition states a simple sufficient condition for Theorem 2.3 to hold and yields Theorem 1.1.
Proposition 2.5. If ξ 1 , ξ 2 , . . . are i.i.d. random vectors in R d with partial sums S k = ξ 1 + · · · + ξ k , k ∈ N, then the following conditions are equivalent: Importantly, the probability in (12) does not depend on the distribution of the increments. This proves the conjecture of Vysotsky and Zaporozhets [VZ17] for general d. Specifying (15) to d = 1 and noting that B(n, 0) = (2n−1)!! we obtain the probability that a symmetric random walk with continuously distributed increments stays positive: This recovers another classical result of Sparre Andersen [And49]. For a simple random walk S k , the formula is different: by the reflection principle, The numbers B(n, k) are called the B-analogs of the (signless) Stirling numbers of the first kind; see the entries A028338 (or A039757 for the signed version) in [Slo]. They satisfy the recurrence relation and are explicitly given by B(n, k) = n i=k 2 n−i i k n i . These numbers were studied in detail by Suter [Sut00]. There is a probabilistic representation of B(n, k): it follows directly from (13) that where δ 1 , . . . , δ n are independent random variables with δ i ∼ Bernoulli( 1 2i ) for 1 ≤ i ≤ n. Geometrically, B(n, k)/(2 n n!) is the kth conic intrinsic volume of the Weyl chamber of type B n ; see Sect. 4.3. 2.3 Type D n . The Coxeter group D n acts on R n by permuting the coordinates in an arbitrary way and by multiplying any even number of coordinates by −1. It is a subgroup of B n of index 2 and the number of its elements is 2 n−1 n!. D n is the symmetry group of the demihypercube constructed from alternation of the regular cube [−1, 1] n .
Remark 2.8. The probability of non-absorption is given by For example, Theorem 2.7 can be applied when ξ 1 , . . . , ξ n are i.i.d. random vectors as in Proposition 2.5. It is easy to show (see (55) below) that for any n ≥ 2, Conv(S 1 , . . . , S n−1 , S n , S * n ) = Conv(S 1 , . . . , S n−1 , S n ) ∪ Conv(S 1 , . . . , S n−1 , S * n ), hence the probabilistic problem corresponding to the symmetry group D n concerns the convex hull of a symmetric random walk allowed to "choose" the sign of its last jump in order to absorb the origin.
The numbers are called the D-analogs of the (signless) Stirling numbers of the first kind; see the entry A039762 in [Slo] for the signed version. It will be shown in Sect. 4.3 that D(n, k)/(2 n−1 n!) is the kth conic intrinsic volume of the Weyl chamber of type D n . Moreover, we have where δ 1 , . . . , δ n are independent random variables with δ i ∼ Bernoulli( 1 2i ) for 1 ≤ i ≤ n − 1 and δ n ∼ Bernoulli( 1 n ).

Direct products of reflection groups.
So far we considered probabilistic problems related to irreducible reflection groups. It is known that a general reflection group can be represented as direct sum of irreducible ones. In this section we study the absorption problem for the joint convex hull of several random walks and/or random walk bridges. The corresponding symmetry groups are the direct products of finite reflection groups.
To be specific, we restrict ourselves to direct products of the form B n1 ×· · ·×B nr containing only groups of the same type, namely B. Here r ∈ N corresponds to the number of random walks and n i , where 1 ≤ i ≤ r, stands for the number of steps in the ith walk. It is straightforward to extend our results to products of the form A n1 × · · · × A nr , which corresponds to joint convex hulls of several random walk bridges, and even to mixed direct products containing groups of all 3 types A, B, D. We omit such extension because it requires more complicated notation.
Since the proof of Theorem 2.9 is based on the same ideas as the proofs in the irreducible cases, but requires complicated notation, it will be presented elsewhere.
Example 2.10 (Type B r 1 : The Wendel formula). Let us consider the particular case n 1 = · · · = n r = 1 that all random walks make just one step. This corresponds to the direct product of r groups Z/2Z, where each factor acts on R by multiplication by ±1.

GAFA CONVEX HULLS OF RANDOM WALKS AND WEYL CHAMBERS 891
The random vectors ξ (1) := ξ for all 2 r choices of the signs. Additionally, we assume that any d of these r random vectors are linearly independent a.s. Then (22), which defines P (k)'s, takes the form (t + 1) r = r k=0 P (k)t k so that P (k) = r k . Theorem 2.9 asserts that Using the recursive property of the Pascal triangle, we obtain This formula is due to Wendel [Wen62], whose proof is essentially based on Schläfli's formula (44) presented below; see [SW08, Section 8.2.1].
The same result can be obtained if one considers the symmetry group A r 1 since its action on R 2r is isomorphic to the action of B r 1 on R r .
Remark 2.11. Although some of our arguments can be extended to other group representations, such extensions do not seem to have a natural probabilistic interpretation. Here is the most meaningful example: by considering the direct product of r dihedral groups, it is possible to find the probability of absorption of the origin by the convex hull of r sides chosen uniformly at random in r regular polygons centred at the origin. We prefer to omit such results here.

2.5
Removing the general position assumptions. As explained above, the general position assumption is essential in our results. Without this assumption, it is still possible to obtain a one-sided bound for the absorption probabilities. Let (ξ 1 , . . . , ξ n ) be a tuple of random vectors in R d that satisfies all the assumptions of any of Theorems 2.1, 2.3, or 2.7. Denote by H n,d the convex hull considered in the respective theorem. Let (ξ 1 , . . . , ξ n ) be any tuple of random vectors in R d that satisfies all the assumptions of the corresponding theorem except the general position one. Put S k = ξ 1 + · · · + ξ k , 1 ≤ k ≤ n, and (S * n ) = ξ 1 + · · · + ξ n−1 − ξ n , and denote by H n,d the convex hull of the respective type.
Note that both H n,d and H n,d are closed, and denote by Int(H n,d ) the interior of H n,d .
Proposition 2.12. For any of the cases A n−1 , B n , D n , we have In particular, this result covers simple random walks on  Proof. Since the absorption probability is distribution-free under the respective general position assumption, we can assume without loss of generality that ξ i = ξ i + εδ i , where ε = 0 and δ 1 , . . . , δ n are random vectors in R d independent of ξ 1 , . . . , ξ n , with the following distribution. In the B n and D n cases, δ 1 , . . . , δ n are i.i.d. standard normal vectors in R d , whereas in the A n−1 case, they are i.i.d. standard normal vectors in R d conditioned on δ 1 + · · · + δ n = 0. The tuple (ξ 1 , . . . , ξ n ) defined in this way satisfies the assumptions of the respective Theorem 2.1, 2.3, or 2.7. Note that the convex hull H n,d is obtained from H n,d by a small random distortion. We have where  (25) and (26) proves (24). Note that the difference in probabilities in (24) can occur because if 0 is on the boundary of H n,d , then even a small distortion possibly gets H n,d aside of 0.
In Sect. 6.3 we will present another proof of Proposition 2.12 which follows our geometric interpretation in terms of intersections of Weyl chambers. It is easy to extend Proposition 2.12 to direct products of reflection groups.

The main formula for the number of regions.
A linear hyperplane arrangement (or simply "arrangement") A is a finite set of distinct hyperplanes in R n that pass through the origin. The literature on hyperplane arrangements [OT92], [Sta07] considers the more general concept of affine hyperplane arrangements (the hyperplanes are not required to pass through the origin) but in the present work we study only the linear case.
The rank of an arrangement A, denoted by rank(A), is the dimension of the linear subspace spanned by the normals to the hyperplanes in A. Equivalently, the rank is the codimension of the intersection of all hyperplanes in the arrangement: The characteristic polynomial χ A (t) of the arrangement A is defined by  (A) the set of open connected components ("regions" or "chambers") of the complement R n \∪ H∈A H of the hyperplanes. The following fundamental result due to Zaslavsky [Zas75] (see also [Sta07, Theorem 2.5]) expresses the number of regions of the arrangement A in terms of its characteristic polynomial: Let A be an arrangement in R n and let L n−d be a linear subspace in R n of Our aim is to find a formula for the number of regions in It is not hard to show, using the fact that R ∩ L n−d is connected in L n−d for every R ∈ R(A), that the regions of the induced arrangement are obtained by intersecting the regions of A with L n−d . Then, clearly, we have 2 In this definition we assume that the linear subspace L n−d is in general position w.r.t. A and that n − d = 1. This ensures that every H ∩ L n−d has codimension 1 in L n−d (by (29)) and that all these hyperplanes are distinct. Indeed, if H1 ∩ L n−d = H2 ∩ L n−d , then both subspaces have dimension n − d − 1 by (29), but, on the other hand, H1 ∩ H2 has dimension d − 2 and hence, H1 ∩ H2 ∩ L n−d has dimension n − d − 2 ≥ 0 by (29), which is a contradiction. In the case that  be the characteristic polynomial of A. Then the characteristic polynomial of A restricted to L n−d is given by Remark 3.2. It is easy to show that a n = 1, a n−1 = #A; see [Sta07,p. 400]. Moreover, the sequence a 0 , . . . , a n is strictly positive [Sta07, Corollary 3.5] and unimodal [Sta07, Lecture 2, Exercise 9 on p. 419]. Let us also prove the identity By the second part of Zaslavsky's theorem [Sta07, Theorem 2.5], for every affine hyperplane arrangement, the number of bounded regions in R(A) is (up to the sign) given by χ A (1) = n k=0 (−1) n−k a k . Since we are dealing only with linear hyperplane arrangements, there are no bounded regions, whence (32).
Proof. If L n−d is a line, then A|L n−d = {{0}} and χ A|Ln−d = t − 1, which is the same expression as in (31) by (32) and since a n = 1.
Suppose in the following that n − d ≥ 2. It follows from (29) that for every subset B ⊂ A, where the rank is in L n−d . Also, as we explained in the footnote, #(B|L n−d ) = #B because L n−d is not a line. Using (27) (in dimension n − d) and then (33) we obtain After noting that by (27) and (30), we obtain the required formula. Now we are ready to state the main result of this section.
where the a k 's are defined by (30) and we set a k = 0 for k / ∈ {0, . . . , n}.

GAFA CONVEX HULLS OF RANDOM WALKS AND WEYL CHAMBERS 895
Proof. By (28) and Lemma 3.1, we have where we used that A|L n−d is an arrangement in dimension n − d. To complete the proof, recall (32).
3.2 Special case: the reflection arrangements. The above results can be applied to the reflection arrangements in R n of the types A n−1 , B n , D n . These arrangements consist of the hyperplanes Proof. The characteristic polynomials of the reflection arrangements are (see Corollary 2.2 on p. 28 and Section 5.1 in [Sta07]) where we have used (8), (13), (18). We stress that A(A n−1 ) is an arrangement in R n , hence its characteristic polynomial has degree n. and The proof will be presented in Sect. 6.3.

Definition of conic intrinsic volumes.
We call a set C ⊂ R n a convex cone if for any x, y ∈ C and a, b > 0 it holds ax + by ∈ C. In the 1940's a spherical counterpart of the Steiner formula was developed in [All48,Her43,San50]. In its modern form (see [SW08, Section 6.5], [San76, Section IV], and [ALMT14,GNP17, MT14]), this formula expresses the size of angular expansions of a closed convex cone C in R n : where θ is a random variable uniformly distributed on the unit sphere S n−1 ⊂ R n and β k,n (·) is the distribution function of a Beta distribution with parameters (n − k)/2 and n/2. Since the functions β 1,n , . . . , β n,n are linearly independent, the formula defines the coefficients υ k (C) uniquely. The quantities υ 0 (C), . . . , υ n (C) are called the conic intrinsic volumes of the cone C. The normalization is chosen so that these quantities do not depend on the dimension of the Euclidean space containing C, and thus conic intrinsic volumes do not change if we consider C as naturally embedded into a space of higher dimension. Note that the kth conic intrinsic volume υ k (C) equals the (k − 1)th spherical intrinsic volume of C ∩ S n−1 considered in [GHS03,SW08].
The conic intrinsic volumes satisfy a version of the Gauss-Bonnet theorem (see, e.g., [SW08, Theorem 6.5.5] or [ALMT14, p. 28]): if C is not a subspace, then The conic analogue of the Crofton formula (see, e.g., [SW08, or [ALMT14,Equation 5.10]) is the following relation: if C is a closed convex cone that is not a subspace, then for every d ∈ {0, . . . , n − 1}, where W n−d is a random (n − d)-dimensional linear subspace in R n chosen w.r.t. the uniform distribution on the Grassmannian.

Characteristic polynomial of linear arrangement and conic intrinsic volumes.
Let A be a linear hyperplane arrangement in R n with characteristic polynomial The next theorem, conjectured by Drton and Klivans [DK10] and proved by Klivans and Swartz [KS11], relates the coefficients of the characteristic polynomial to the conic intrinsic volumes of the regions of the arrangement. We will give a completely different (and very short) proof of this theorem. Our approach was already extended by Schneider [Sch, Theorem 1.2]. A generalization of the Klivans-Swartz formula [KS11] was first considered by Amelunxen and Lotz [AL, Section 6], whose work appeared after the first version of our paper. h 0 (R) = a 0 + a 2 + . . . . (40)) we obtain that for all k ∈ {0, . . . , n}, This completes the proof.

Conic intrinsic volumes of the Weyl chambers.
The Weyl chambers of type A n−1 , B n , D n are the following convex cones in R n : Each Weyl chamber C = C(G) is a fundamental domain for the corresponding reflection group G = A n−1 , B n or D n . This means that all cones of the form gC, g ∈ G, which are also called Weyl chambers without any risk of confusion, are disjoint and that ∪ g∈G gC = R n holds true.

Random arrangements of hyperplanes in general position. Let
A be a linear arrangement in R n consisting of m ≥ n hyperplanes in general position; in our terminology, this means that R n is in general position w.r.t. A; see (29). By (27) we have that Applying (28) and using that the alternating binomial coefficients add up to zero, we get hence by the recursive property of the Pascal triangle, . . , X m be i.i.d. random vectors on the unit sphere S n−1 such that their common distribution is centrally symmetric and assigns no mass to any (n − 2)dimensional great subsphere. The hyperplanes X ⊥ 1 , . . . , X ⊥ m , which are in general position a.s., divide R n into C(m, n) random cones. We choose one of these cones uniformly at random to obtain the random Schläfli cone C m in R n introduced by Hug and Schneider in [HS16].
The next result of [HS16] (given there with a slightly different notation) calculates the expected intrinsic volumes of a random Schläfli cone. This theorem easily follows from Theorem 4.1.

Theorem 4.3. For any random Schläfli cone
Proof. Let A be the arrangement consisting of the hyperplanes X ⊥ 1 , . . . , X ⊥ m . We have On the other hand, it follows from Theorem 4.1 and (43) that which completes the proof.

Asymptotic results
In this section we use the exact expressions of Sect. 2 to study the asymptotic behavior of the probability that the convex hull of a symmetric random walk or of a random walk bridge absorbs the origin. The cases A n−1 , B n and D n are very similar. We work in the setting of Theorems 2.1, 2.3, 2.7, which assume the exchangeability of increments, the assumption of general position, and in the cases B n and D n , the symmetry of the distribution of increments. Recall that H n,d ⊂ R d denotes the convex hull considered in any of these theorems.

Asymptotics in constant dimension.
The following theorem gives the asymptotics of the non-absorption probability in the case that the dimension d is fixed and the number of steps n tends to infinity. In the case d = 2 this result was obtained by Vysotsky and Zaporozhets [VZ17]. We write x n ∼ n→∞ y n if lim n→∞ x n /y n = 1.
πn , in the B n and D n cases. Proof. In the A n−1 case, we can use the well-known asymptotics of the Stirling numbers [Wil93]: for a fixed k ∈ N, Substituting this formula into the statement of Theorem 2.1 (see also (10)) and noting that the term n d dominates all other terms, we obtain In the B n case, the definition of B(n, k) given in (13) yields where, in order to pass from the first line to the second one, one shows (by an omitted standard argument) that the contribution of all terms in the sum where at least two indices i m and i l are equal is o((log n) k ). Substituting the above asymptotics for B(n, k) into the statement of Theorem 2.3 (see also (15)), and noting that the term B(n, d − 1) dominates all other terms, we obtain Using the asymptotics of B(n, d − 1) and the Stirling formula, we obtain The computation for the D n case is similar to the one for the B n case and yields the same result.

High-dimensional asymptotics: central limit theorem.
Consider the convex hull H n,d of a symmetric random walk (or any random walk bridge) of length n in a high dimension d. It is clear that if n is sufficiently small, then the absorption probability should be close to 0, whereas for sufficiently large n the absorption probability should be close to 1. Hence, at some value of n (which is a function of d) there should be a phase transition from non-absorption to absorption. This transition was studied by Eldan [Eld14] and in the subsequent paper by Tikhomirov and Youssef [TY17]. In this section we provide a precise description of the location of this phase transition. It will be convenient for us to make d = d(n) a function of n rather than considering n as a function of d. The next theorem shows, in particular, that the absorption probabilities P[0 ∈ H n,d(n) ] exhibit a phase transition at d(n) ≈ 1 2 log n for symmetric random walks and d(n) ≈ log n for random walk bridges.
Proof. Consider the case A n−1 first. By Theorem 2.1 (see (10)), we have On the other hand, the Stirling numbers of the first kind are unimodal (in k) being the (signed) coefficients of the characteristic polynomial of a hyperplane arrangement; see Remark 3.2 and (34). Combining this fact with (47) and the unimodality of the standard normal density, we see that the mode m n of the sequence n k satisfies m n = u log n + o( log n).
Hence if a < 0, then n k are monotone increasing in k for k < d(n), and thus By the Goncharov CLT (47), this implies the required (45) for a < 0. The proof for a > 0 follows analogously by considering the complement probabilities P[0 ∈ H n,d(n) ] and using the monotonicity of n k for k > d(n). Finally, the case a = 0 is easily treated using the continuity of the standard normal density. This completes the proof of Theorem 5.2 in the case A n−1 .
We now turn to the case B n . We will use the powerful theory of mod-Poisson convergence developed in [FMN16,KN10]. Once established, the mod-Poisson convergence yields many limit theorems besides the CLT.
Let X n be an integer-valued random variable with the distribution P[X n = k] = 1 2 n n! B(n, k), k = 0, . . . , n.
Note that the probabilities indeed sum up to one by (13) with t = 1. We claim that X n satisfies a central limit theorem of the form where N(0, 1) is the standard normal law. This is analogous to (47). Denote by (x) n = x(x + 1) . . . (x + n − 1) the rising factorial. By the definition of B(n, k) given in (13), the moment generating function of X n is ( 1 2 e z ) n .

GAFA CONVEX HULLS OF RANDOM WALKS AND WEYL CHAMBERS 903
Recall that (x) n ∼ n x Γ(n)/Γ(x) as n → ∞. This holds locally uniformly in x ∈ C and follows from the Weierstrass infinite product formula for 1/Γ(x). Using this asymptotics and the Stirling formula, we obtain the following.
Lemma 5.3. For the sequence X n defined in (50), locally uniformly on C we have The denominator e ( 1 2 log n)(e z −1) on the left-hand side is the moment generating function of a Poisson distribution with parameter 1 2 log n. Hence (52) states that X n converges in the mod-Poisson sense. This implies the CLT (51) by the general theory of mod-Poisson convergence; see [KN10, Proposition 2.4, Part (2)].
The rest of the proof is completely analogous to the case A n−1 . The probability mass function of X n is unimodal by Remark 3.2 and (35), and its mode satisfies (48) (with u = 1 2 ) by the established CLT (51). By Theorem 2.3 (see (15)) combined with the definition of X n given in (50) and the unimodality of B(n, k), we see that if a < 0, then This is analogous to (49) and proves the required (45) in the B n case for a < 0. The case a ≥ 0 is covered as above.
The D n case is completely analogous to the B n case and yields the same asymptotics.
Corollary 5.4. It holds Proof. The third case follows by taking a = 0 in Theorem 5.2. The other two cases also follow from Theorem 5.2 since the probability P[0 / ∈ H n,d ] is increasing in d if we restrict d to be even or odd; see Theorems 2.1, 2.3, 2.7.

High-dimensional asymptotics: large deviations.
In this section we give the asymptotics for the absorption (non-absorption) probabilities in regions of large deviations, where the random walk or bridge makes too few (respectively, too many) steps compared to a typical mode described in the previous section. The first results of this kind were obtained by Eldan [Eld14], who proved that for some constants 0 < c 1 < c 2 , non-absorption (respectively, absorption) occurs with a high probability provided that n < e c1d/ log d (respectively, n > e c2d log d ). Tikhomirov and Youssef [TY17] removed the log d factor and replaced the bounds above by n < e c1d and n > e c2d , respectively. A n−1 , B n , D n : Proofs of Theorems 2.1, 2.3, and 2.7.

Reflection groups
We identify the elements of the Coxeter groups A n−1 , B n , and D n with orthogonal transformations g : R n → R n . The Weyl chambers C(A n−1 ), C(B n ), and C(D n ) are respective fundamental domains for the actions of these groups. The action of A n−1 leaves the hyperplane L given by the equation x 1 + · · · + x n = 0 invariant, hence C(A n−1 ) ∩ L is a fundamental domain for the action of A n−1 restricted on L.
Let ξ 1 , . . . , ξ n be random vectors with values in R d (written as columns), and let A be the random d×n-matrix with columns ξ 1 , . . . , ξ n . We regard A as a random linear operator A : R n → R d . The kernel Ker A of this operator is a random linear subspace of R n . Recall that H n,d is the common notation for the convex hulls considered in Theorems 2.1, 2.3, 2.7.
Lemma 6.1. Let G be any of the Coxeter groups A n−1 , B n , D n , and let C denote the respective domain C(A n−1 ) ∩ L, C(B n ), or C(D n ). Suppose that the tuple (ξ 1 , . . . , ξ n ) satisfies all the assumptions of the respective Theorem 2.1, 2.3, or 2.7 except the one on general position. Then for every g ∈ G, Proof. We are interested in the probability of the event Denote by e 1 , . . . , e n the standard basis of R n , and recall that S 1 = ξ 1 , S 2 = ξ 1 + ξ 2 , . . . , S n = ξ 1 + · · · + ξ n .
Type A n−1 . The elements of A n−1 are the orthogonal transformations of the form g σ : R n → R n , where σ ∈ Sym(n) is a permutation on n elements, and g σ (e k ) = e σ(k) , k = 1, . . . , n.
To complete the proof of Lemma 6.1, we need to argue that The left-hand side is a subset of the right-hand side by definition of the convex hull. To see the converse inclusion, consider any convex combination x = α 1 S 1 + · · · + α n−2 S n−2 + α n−1 S n−1 + α n S n + α * n S * n .
If α n ≥ α * n , then by the identity S * n = 2S n−1 − S n , we obtain x = α 1 S 1 + · · · + α n−2 S n−2 + (α n−1 + 2α * n )S n−1 + (α n − α * n )S n , which represents x as a convex combination of S 1 , . . . , S n−1 , S n . Indeed, the sum of the coefficients did not change, and hence equals one. This also ensures that the coefficients do not exceed one since all of them are non-negative. Similarly, if α n ≤ α * n , we obtain representation of x as a convex combination of S 1 , . . . , S n−1 , S * n .
We are now ready to complete the proofs of Theorems 2.1, 2.3, and 2.7. Applying Lemma 6.1 to all g ∈ G and taking the arithmetic mean, we obtain where the random variable counts the number of chambers of the form gC, g ∈ G, intersected by Ker A.
In the A n−1 case, N equals the number of closed Weyl chambers of type A n−1 (non-trivially) intersected by L ∩ Ker A, as readily seen from the equation Ker A ∩ (gC) = (L ∩ Ker A) ∩ g(C(A n−1 )). In the B n and D n cases, N equals the number of closed Weyl chambers of the respective type that have a non-trivial intersection with Ker A.
It remains to use the following lemmas, whose proof is postponed for a moment. Proof of Lemma 6.2. We use β 1 , . . . , β n as coordinates on R n . Recall that Ker A is the set of solutions to the system of effectively d linear equations β 1 ξ 1 +· · ·+β n ξ n = 0. Define the linear function T : R n → L that maps (β 1 , . . . , β n ) to (β 1 − b, . . . , β n − b), where b = 1 n (β 1 + · · · + β n ). Note that T (Ker A) = L ∩ Ker A a.s. by ξ 1 + · · · + ξ n = 0 a.s. Since Ker T = L ⊥ = {(a, . . . , a) : a ∈ R} belongs to Ker A a.s., the kernel of the restriction T | Ker A also is L ⊥ a.s. Hence where the last equality holds since ξ 1 , . . . , ξ d are linearly independent a.s., which in turn easily follows from the a.s. linear independence of S 1 , . . . , S d given by the general position assumption of Theorem 2.1. Therefore the rank of A equals d with probability 1.
Let us prove that L ∩ Ker A is in general position w.r.t. A(A n−1 ). Take a linear subspace K ⊂ R n of dimension k that can be represented as the intersection of hyperplanes from A(A n−1 ), i.e., hyperplanes of the form β i = β j , 1 ≤ i < j ≤ n. According to (29), we need to show that Similarly to the above, we have T (K ∩ Ker A) = K ∩ L ∩ Ker A a.s. and the kernel of the restriction T | K∩Ker A a.s. is the one-dimensional linear space L ⊥ . Thus it suffices to show that The linear subspace K is given by a system of equations of the following type: the variables β 1 , . . . , β n are decomposed into k non-empty groups and required to be equal inside each group. Since (ξ 1 , . . . , ξ n ) is an exchangeable tuple and we can apply a suitable transformation from the group A n−1 , it can be assumed without loss of generality that K is given by the equations γ 1 := β 1 = · · · = β i1 , γ 2 := β i1+1 = · · · = β i2 , . . . , γ k := β ik−1+1 = · · · = β n , for some 1 ≤ i 1 < · · · < i k−1 < i k := n.
Let now k < d + 1. Take some linear subspace K 1 ⊃ K that can be represented as the intersection of hyperplanes from the arrangement A(A n−1 ) and satisfies dim K 1 = d + 1. Then apply the above to get dim(K 1 ∩ Ker A) = d + 1 − d = 1 a.s. This yields dim(K ∩ Ker A) ≤ 1 a.s., but since L ⊥ ⊂ K ∩ Ker A a.s., we in fact have dim(K ∩ Ker A) = 1 a.s., thus completing the proof.
Proof of Lemma 6.3. Consider the B n case first. By the general position assumption imposed in Theorem 2.3, the vectors S 1 , . . . , S d are linearly independent a.s. Hence ξ 1 , . . . , ξ d are linearly independent a.s. and the rank of the matrix A equals d with probability 1. Then the codimension of Ker A equals d a.s.
Letting β 1 , . . . , β n denote the coordinates on R n , observe that Ker A is given by β 1 ξ 1 + · · · + β n ξ n = 0. To prove that Ker A a.s. is in general position w.r.t. A(B n ), take a linear subspace K ⊂ R n of dimension k that can be represented as the intersection of hyperplanes from A(B n ), that is, hyperplanes of the form According to the definition of general position (see (29)), we have to show that The linear subspace K is given by a system of equations of the following form. The variables β 1 , . . . , β n are decomposed into k + 1 distinguishable groups, all of which must be non-empty except the last one. All variables in the last group are required to be 0. For the remaining variables, there is a unique choice of signs, which multiplies each variable by +1 or −1, such that the sign-changed variables are equal inside every group except the (k + 1)st one.
Using the partial sums, this can be written as Let first k = dim K ≥ d. Then with probability 1, the rank of this system of equations is maximal (namely, d) since S i1 , . . . , S id and hence, S i1 , S i2 −S i1 , . . . , S id − S id−1 , are linearly independent a.s. by our general position assumption. Then K ∩ Ker A, the space of solutions of the system, has dimension k − d a.s. as required.
Let now k < d. Take some linear subspace K 1 ⊃ K representable as the intersection of hyperplanes from the arrangement A(B n ) and satisfying dim K 1 = d. Applying the above to K 1 , we get dim(K 1 ∩ Ker A) = d − d = 0 a.s., which yields dim(K ∩ Ker A) = 0 a.s., completing the proof in the B n case.
The D n case can be treated similarly, and we highlight only the main differences. Let K ⊂ R n be a linear subspace of dimension k that can be represented as the intersection of hyperplanes of the form β i = ±β j , 1 ≤ i < j ≤ n. Then K has exactly the same form as in the B n case, but since the arrangement A(D n ) does not include the hyperplanes β i = 0, the last group of variables (that are required to be 0) cannot contain exactly one element. Applying an appropriate transformation from the group D n allows to change only even number of signs, therefore we can assume that K either has the same form as in the B n case, or is given by for some 1 ≤ i 1 < · · · < i k−1 < i k := n. In the former case, the same argument as in the B n case applies. In the latter case, K ∩ Ker A is given (inside the linear space K) by where we recall that S * n = S n−1 − ξ n . From now on, we can apply the same argument as in the B n case, but with S n replaced by S * n .
Proof of Proposition 2.5 and Theorem 1.1. It suffices to prove the equivalence of (i), (ii) and (iii) in Proposition 2.5, because then Theorem 1.1 follows as a particular case of Theorem 2.3.
Proof of (ii) ⇒ (iii). Let H 0 be given by the equation Proof of (iii) ⇒ (i). The vectors S i1 , . . . , S id are linearly dependent if and only if S ik can be linearly expressed through S i1 , . . . , S ik−1 for some 2 ≤ k ≤ d, or if S i1 = 0. The latter event has probability zero by (iii). To prove that the former event also has probability 0, denote by lin(y 1 , . . . , y k−1 ) the linear subspace spanned by vectors y 1 , . . . , y k−1 ∈ R d . Then since the integrand is 0 by (iii). Hence the probability that S i1 , . . . , S id are linearly dependent is 0.

Non-general position: Proofs of Lemma 3.5 and Proposition 2.12.
Proof of Lemma 3.5. Let us prove (36), that is Since 0 / ∈ R, the assumption R ∩ L n−d = ∅ clearly implies thatR ∩ L n−d = {0}. To prove the other inclusion in (58), we assume, by contraposition, thatR ∩ L n−d = {0} but R ∩ L n−d = ∅. Note thatR is a closed polyhedral cone, that is an intersection of finitely many closed half-spaces whose boundaries are hyperplanes passing through the origin.
Given some face F ofR, we denote by lin F its linear hull and by m ∈ {0, . . . , n} the dimension of lin F (we assume thatR itself is a face). The relative interior relint F of the face F is defined as the interior of F taken w.r.t. the linear hull lin F as the ambient space. It is known thatR is the disjoint union of the relative interiors of its faces. Hence, there is a face F = {0} ofR such that L n−d ∩ relint F = ∅. Sincē R is a cone, the dimension of F is at least one. Note that F =R by the assumption that R ∩ L n−d = ∅, hence dim F = m / ∈ {0, n}. Since L n−d is in general position w.r.t. A, the dimension of the linear space V 0 := L n−d ∩ lin F is max(m − d, 0). In fact, it equals m − d = 0, because otherwise we would have L n−d ∩ lin F = {0}, which is a contradiction. Thus, we can construct linear spaces V 1 and V 2 such that lin F = V 0 ⊕ V 1 , L n−d = V 0 ⊕ V 2 , and V 0 ⊕ V 1 ⊕ V 2 = R n . We then have dim V 1 = d and dim V 2 = n − m = 0. Note that V 1 and V 2 can be taken so that V 0 ⊥V 1 and V 0 ⊥V 2 but we do not necessarily have V 1 ⊥V 2 .
Take some x ∈ L n−d ∩ relint F . The support (or tangent) cone ofR at its face F is defined as A(F ) = pos(R − x) = {y ∈ R n : ∃ε > 0 such that x + εy ∈R}, where pos(M ), the positive hull of M , is the minimal convex cone containing the set M . It is known that A(F ) is a closed convex cone containing lin F and not depending on the choice of x ∈ relint F . This assumption on x also ensures that there is a δ > 0 such that B δ (x) ∩R = B δ (x) ∩ (x + A(F )).
Let z ∈ R be some point from the interior ofR. Write z −x = v 0 +v 1 +v 2 ∈ A(F ) with v i ∈ V i , i = 0, 1, 2. Since −(v 0 + v 1 ) ∈ lin F ⊂ A(F ), we have v 2 ∈ A(F ). In fact, the same argument applies to any point in a sufficiently small ball around z. Observe that v 2 is the projection of z − x onto V 2 along V 0 ⊕ V 1 , and that the projection of the ball around z − x covers some set of the form B r (v 2 ) ∩ V 2 , where B r (v 2 ) is the ball of radius r > 0 around v 2 . This proves that B r (v 2 ) ∩ V 2 ⊂ A(F ), but since V 0 ⊕ V 1 = lin F ⊂ A(F ), we even have B r (v 2 ) ⊂ A(F ) for some 0 < r ≤ r by the convex cone property of A(F ). Then, by the same property, B εr (εv 2 ) ⊂ B δ (0) ∩ A(F ) for all sufficiently small ε > 0. Hence, using (59), we get B εr (x + εv 2 ) ⊂ B δ (x) ∩ (x + A(F )) ⊂R. This implies that x + εv 2 is in the interior ofR, which is a contradiction because we also have x ∈ L n−d , v 2 ∈ L n−d and, consequently, x + εv 2 ∈ L n−d .
Now we prove (37); the proof of (38) is analogous. So, we need to prove that   Recall that (ξ 1 , . . . , ξ n ) is a symmetrically exchangeable tuple of random vectors in R d and H n,d = Conv(S 1 , . . . , S n ) is the convex hull of the partial sums S k = ξ 1 + · · · + ξ k . Let A be the d × n-matrix with columns ξ 1 , . . . , ξ n . By Lemma 6.1, where N is the numbers of closed Weyl chambers of type B n (non-trivially) intersected by Ker A : We imposed no general position assumption on (ξ 1 , . . . , ξ n ) and we cannot claim that Ker A is in general position w.r.t. the arrangement A(B n ). In particular, the random variable N need not be a constant a.s. Moreover, we do not even known the exact codimension of Ker A , but we can claim that it is at most min(d, n). Let F ⊂ Ker A be any random linear subspace of R n of a.s. codimension d. For example, we may define it as follows. Put κ = min(d − codim(Ker A ), n) and take F = By Lemma 3.5, we haveÑ ≥ N a.s. for N defined by (57) with G = B n . By (56), the claim follows.

Open questions
Our results, except the estimates of Theorem 5.7 and Proposition 2.12, do not apply to simple random walks on the lattice Z d . The next problem does not seem trivial even for d = 2.
Problem 7.1. Let S 1 , . . . , S n be a simple random walk on Z d starting at the origin. Compute exactly the probability that Conv(S 1 , . . . , S n ) contains the origin. Compute exactly the conditional probability that Conv(S 1 , . . . , S n−1 ) contains the origin given that S n = 0.
Problem 7.2. Prove analogues of Theorems 5.2 and 5.5 for the simple random walks (and bridges) on Z d .