Open Access
2020 The coin-turning walk and its scaling limit
János Engländer, Stanislav Volkov, Zhenhua Wang
Electron. J. Probab. 25: 1-38 (2020). DOI: 10.1214/19-EJP406


Let $S$ be the random walk obtained from “coin turning” with some sequence $\{p_{n}\}_{n\ge 2}$, 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$\cdot n^{-1}$ (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 $n$th step of the walk.


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János Engländer. Stanislav Volkov. Zhenhua Wang. "The coin-turning walk and its scaling limit." Electron. J. Probab. 25 1 - 38, 2020.


Received: 29 April 2019; Accepted: 22 December 2019; Published: 2020
First available in Project Euclid: 8 January 2020

zbMATH: 07149390
MathSciNet: MR4053903
Digital Object Identifier: 10.1214/19-EJP406

Primary: 60F05 , 60G50 , 60J10

Keywords: coin-turning , cooling dynamics , heating dynamics , invariance principle , Random walk , Scaling limit , time-inhomogeneous Markov-process , zigzag process

Vol.25 • 2020
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