## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 25 (2020), paper no. 3, 38 pp.

### The coin-turning walk and its scaling limit

János Engländer, Stanislav Volkov, and Zhenhua Wang

#### Abstract

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.

#### Article information

**Source**

Electron. J. Probab., Volume 25 (2020), paper no. 3, 38 pp.

**Dates**

Received: 29 April 2019

Accepted: 22 December 2019

First available in Project Euclid: 8 January 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1578452592

**Digital Object Identifier**

doi:10.1214/19-EJP406

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

coin-turning random walk scaling limit time-inhomogeneous Markov-process Invariance Principle cooling dynamics heating dynamics zigzag process

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Engländer, János; Volkov, Stanislav; Wang, Zhenhua. The coin-turning walk and its scaling limit. Electron. J. Probab. 25 (2020), paper no. 3, 38 pp. doi:10.1214/19-EJP406. https://projecteuclid.org/euclid.ejp/1578452592