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

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posted on 2023-06-09, 07:17 authored by Zakhar Kabluchko, Vladislav VysotskiyVladislav Vysotskiy, Dmitry Zaporozhets
Consider a random walk Si = ?1 + . . . + ?i , i ? N, whose increments ?1, ?2, . . . are independent identically distributed random vectors in R d such that ?1 has the same law as -?1 and P[?1 ? H] = 0 for every affine hyperplane H ? R d . Our main result is the distribution-free formula [see published version for formula] where the B(k, j)’s are defined by their generating function (t + 1)(t + 3). . .(t + 2k - 1) = Pk j=0 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 Bn that are not intersected by a generic linear subspace L ? 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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Bernoulli

ISSN

1350-7265

Publisher

Bernoulli Society for Mathematical Statistics and Probability

Issue

1

Volume

25

Page range

521-548

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-12-20

First Open Access (FOA) Date

2018-04-27

First Compliant Deposit (FCD) Date

2017-12-20

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