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

TL;DR

This paper generalizes the classical arcsine law to multidimensional random walks, providing a distribution-free formula for the expected number of positive terms and relating it to geometric properties of Weyl chambers.

Contribution

It introduces a multidimensional analogue of the arcsine law for the count of positive terms in a random walk, extending classical results to higher dimensions with a geometric interpretation.

Findings

Derived a distribution-free formula for the expected number of positive terms in multidimensional random walks.

Connected the expected count to the number of certain faces in Weyl chambers.

Extended results to random bridges without symmetry assumptions.

Abstract

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.