Abstract. We give an explicit formula for the probability that the convex hull of an n-step random walk in R d 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 B n intersected by a generic linear subspace of R n of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type B n . 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 A n−1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type D n , 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 B 1 × · · · × B 1 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.

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that $p_N \le c N^{-1/4}$ for lattice walks and for upper exponential walks, that are the walks such that $Law (S_1 | S_1>0)$ is an exponential distribution.Comment: Theorems 2 and 3 were restated and merged into one theorem; a new lemma (Lemma 1) added; Lemma 3 and Remark 1 were restated and merged into Proposition 1; the proof of Lemma 3 is reworked. The paper is accepted to SPA

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.