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

Abstract

Consider a random walk $S_{i}=\xi_{1}+\cdots+\xi_{i}$, $i\in\mathbb{N}$, 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

\[\mathbb{E}\bigg[\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\mathbb{1}_{\{0\notin\operatorname{Conv}(S_{i_{1}},\ldots,S_{i_{k}})\}}\bigg]=2\binom{n}{k}\frac{B(k,d-1)+B(k,d-3)+\cdots}{2^{k}k!},\] 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.