A multidimensional analogue of the arcsine law for the number of positive terms in a random walk

Consider a random walk Si = ξ1 + . . . + ξi , i ∈ N, whose increments ξ1, ξ2, . . . are independent identically distributed random vectors in R d such that ξ1 has the same law as −ξ1 and P[ξ1 ∈ H] = 0 for every affine hyperplane H ⊂ R d . Our main result is the distribution-free formula [see published version for formula] where the B(k, j)’s are defined by their generating function (t + 1)(t + 3). . .(t + 2k − 1) = Pk j=0 B(k, j)t j . The expected number of k-tuples above admits the following geometric interpretation: it is the expected number of k-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type Bn that are not intersected by a generic linear subspace L ⊂ R n of codimension d. The case d = 1 turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required