The coin-turning walk and its scaling limit
(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.
(Less)
- author
- Engländer, János ; Volkov, Stanislav LU and Wang, Zhenhua
- organization
- publishing date
- 2020
- 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}}, }