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

Consider a random walk $S_i= \xi_1+\ldots+\xi_i$, $i\in\mathbb N$, whose increments $\xi_1,\xi_2,\ldots$ are independent identically distributed random vectors in $\mathbb R^d$ such that $\xi_1$ has the same law as $-\xi_1$ and $\mathbb P[\xi_1\in H] = 0$ for every affine hyperplane $H\subset \mathbb R^d$. Our main result is the distribution-free formula $$ \mathbb E \left[\sum_{1\leq i_1 < \ldots < i_k\leq n} 1_{\{0\notin \text{conv}(S_{i_1},\ldots, S_{i_k})\}}\right] = 2 \binom n k \frac {B(k, d-1) + B(k, d-3) +\ldots} {2^k k!}, $$ where the $B(k,j)$'s are defined by their generating function $$ (t+1) (t+3) \ldots (t+2k-1) = \sum_{j=0}^{k} B(k,j) t^j. $$ 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 $B_n$ that are not intersected by a generic linear subspace $L\subset \mathbb R^n$ 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.