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. 
Zakhar Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets
math.PR math.CO math.MG