Electronic Journal of Probability

The coin-turning walk and its scaling limit

János Engländer, Stanislav Volkov, and Zhenhua Wang

Full-text: Open access

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


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References

  • [1] Arratia, R.; Barbour, A. D.; Tavaré, S. The Poisson-Dirichlet distribution and the scale-invariant Poisson process. Combin. Probab. Comput. 8, (1999), no. 5, 407–416.
  • [2] Benaïm, M.; Bouguet, F.; Cloez, B. Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories. Ann. Appl. Probab. 27, (2017), no. 5, 3004–3049.
  • [3] Bouguet, F.; Cloez, B. Fluctuations of the empirical measure of freezing Markov chains. Electron. J. Probab. 23, (2018), Paper No. 2.
  • [4] Breiman, L. Probability. Corrected reprint of the 1968 original. Classics in Applied Mathematics, 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  • [5] Davydov, Ju. A. The invariance principle for stationary processes. (Russian) Teor. Verojatnost. i Primenen. 15, (1970), 498–509.
  • [6] Davydov, Ju. A. Mixing conditions for Markov chains. (Russian) Teor. Verojatnost. i Primenen. 18, (1973), 321–338.
  • [7] Drogin, R. An invariance principle for martingales. Ann. Math. Statist. 43, (1972), 602–620.
  • [8] Engländer, J.; Volkov, S. Turning a coin over instead of tossing it. J. Theor. Probab. 31, (2018), 1097–1118.
  • [9] Kallenberg, O. Random measures, theory and applications. Probability Theory and Stochastic Modeling, 77. Springer, Cham, 2017.
  • [10] Karatzas, I.; Shreve, S. E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991.
  • [11] Pólya, G.; Szegő, G. Problems and theorems in analysis. I. Series, integral calculus, theory of functions. Translated from the German by Dorothee Aeppli. Reprint of the 1978 English translation. Classics in Mathematics. Springer-Verlag, Berlin, 1998.