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The coin-turning walk and its scaling limit

Engländer, János ; Volkov, Stanislav LU orcid and Wang, Zhenhua (2020) In Electronic Journal of Probability 25.
Abstract

Let S be the random walk obtained from “coin turning” with some sequence {pn}n≥2, as introduced in [8]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n−1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a... (More)

Let S be the random walk obtained from “coin turning” with some sequence {pn}n≥2, as introduced in [8]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n−1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Coin-turning, Cooling dynamics, Heating dynamics, Invariance Principle, Random walk, Scaling limit, Time-inhomogeneous Markov-process, Zigzag process
in
Electronic Journal of Probability
volume
25
article number
3
publisher
UNIV WASHINGTON, DEPT MATHEMATICS
external identifiers
  • scopus:85078354320
ISSN
1083-6489
DOI
10.1214/19-EJP406
language
English
LU publication?
yes
id
07994c37-dd3b-4c50-a2de-52b19c11ea3d
date added to LUP
2020-02-10 13:09:29
date last changed
2022-04-18 20:26:19
@article{07994c37-dd3b-4c50-a2de-52b19c11ea3d,
  abstract     = {{<p>Let S be the random walk obtained from “coin turning” with some sequence {p<sub>n</sub>}<sub>n≥2</sub>, as introduced in [8]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n<sup>−1</sup> (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.</p>}},
  author       = {{Engländer, János and Volkov, Stanislav and Wang, Zhenhua}},
  issn         = {{1083-6489}},
  keywords     = {{Coin-turning; Cooling dynamics; Heating dynamics; Invariance Principle; Random walk; Scaling limit; Time-inhomogeneous Markov-process; Zigzag process}},
  language     = {{eng}},
  publisher    = {{UNIV WASHINGTON, DEPT MATHEMATICS}},
  series       = {{Electronic Journal of Probability}},
  title        = {{The coin-turning walk and its scaling limit}},
  url          = {{http://dx.doi.org/10.1214/19-EJP406}},
  doi          = {{10.1214/19-EJP406}},
  volume       = {{25}},
  year         = {{2020}},
}