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

Abstract

Consider a random walk $S_i={\xi }_1+\dots +{\xi }_i$, $i\in \mathbb{N}$, whose increments ${\xi }_1,{\xi }_2,\dots$ 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 $$\mathbb{E}\left[\sum _{1\le i_1<\dots <i_k\le n}{1}_{\{0\notin \text{conv}(S_{i_1},\dots ,S_{i_k})\}}\right]=2\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{B(k,d-1)+B(k,d-3)+\dots }{2^{k}k!},$$ where the $B(k,j)$'s are defined by their generating function $$(t+1)(t+3)\dots (t+2k-1)=\sum _{j=0}^{k}B(k,j)t^j.$$ 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.

arXiv comment25 pages, no figures