Bernoulli

  • Bernoulli
  • Volume 25, Number 1 (2019), 521-548.

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

Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets

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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.

Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 521-548.

Dates
Received: November 2016
Revised: August 2017
First available in Project Euclid: 12 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1544605255

Digital Object Identifier
doi:10.3150/17-BEJ996

Mathematical Reviews number (MathSciNet)
MR3892328

Zentralblatt MATH identifier
07007216

Keywords
absorption probability arcsine law convex cone convex hull distribution-free probability finite reflection group hyperplane arrangement random linear subspace random walk random walk bridge Weyl chamber

Citation

Kabluchko, Zakhar; Vysotsky, Vladislav; Zaporozhets, Dmitry. A multidimensional analogue of the arcsine law for the number of positive terms in a random walk. Bernoulli 25 (2019), no. 1, 521--548. doi:10.3150/17-BEJ996. https://projecteuclid.org/euclid.bj/1544605255


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