Final text.pdf (261.23 kB)
A multidimensional analogue of the arcsine law for the number of positive terms in a random walk
journal contribution
posted on 2023-06-09, 07:17 authored by Zakhar Kabluchko, Vladislav VysotskiyVladislav Vysotskiy, Dmitry ZaporozhetsConsider 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
BernoulliISSN
1350-7265Publisher
Bernoulli Society for Mathematical Statistics and ProbabilityExternal DOI
Issue
1Volume
25Page range
521-548Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2017-12-20First Open Access (FOA) Date
2018-04-27First Compliant Deposit (FCD) Date
2017-12-20Usage metrics
Categories
No categories selectedKeywords
Licence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC