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Arcsine laws for random walks generated from random permutations with applications to genomics

Part of: Markov processes Combinatorics Combinatorial probability

Published online by Cambridge University Press:  22 November 2021

Xiao Fang ,
Han L. Gan ,
Susan Holmes ,
Haiyan Huang ,
Erol Peköz ,
Adrian Röllin Open the ORCID record for Adrian Röllin [Opens in a new window]  and
Wenpin Tang
Xiao Fang*
Affiliation:
The Chinese University of Hong Kong
Han L. Gan*
Affiliation:
Northwestern University
Susan Holmes*
Affiliation:
Stanford University
Haiyan Huang*
Affiliation:
University of California, Berkeley
Erol Peköz*
Affiliation:
Boston University
Adrian Röllin*
Affiliation:
National University of Singapore
Wenpin Tang*
Affiliation:
Columbia University
*
*Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: xfang@sta.cuhk.edu.hk
**Postal address: University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand. Email: han.gan@waikato.ac.nz
***Postal address: Department of Statistics, 390 Jane Stanford Way, Stanford University, Stanford, CA 94305-4020. Email: susan@stat.stanford.edu
****Postal address: Department of Statistics, University of California, Berkeley, 367 Evans Hall, Berkeley, CA 94720-3860. Email: hhuang@stat.berkeley.edu
*****Postal address: Boston University, Questrom School of Business, Rafik B. Hariri Building, 595 Commonwealth Avenue, Boston, MA 02215. Email: pekoz@bu.edu
******Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546. Email: adrian.roellin@nus.edu.sg
*******Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 500 W. 120th Street #315, New York, NY 10027. Email: wt2319@columbia.edu
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Abstract

A classical result for the simple symmetric random walk with 2n steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows(q) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step 2n for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Stein’s method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows(q) permutation which is of independent interest.

Keywords

Brownian motionLévy statisticsMallows permutationStein’s methodstrong embeddinguniform permutation

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60J65: Brownian motion 05A05: Permutations, words, matrices
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 851 - 867
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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