Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-29T05:14:36.667Z Has data issue: false hasContentIssue false
  • English
  • Français

Variations of the elephant random walk

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  16 September 2021

Allan Gut  and
Ulrich Stadtmüller Open the ORCID record for Ulrich Stadtmüller [Opens in a new window]
Allan Gut*
Affiliation:
Uppsala University
Ulrich Stadtmüller*
Affiliation:
Ulm University
*
*Postal address: Uppsala University, Department of Mathematics, Box 480, SE-751 06 Uppsala, Sweden. Email address: allan.gut@math.uu.se
**Postal address: Ulm University, Department of Number and Probability Theory, 89069 Ulm, Germany. Email address: ulrich.stadtmueller@uni-ulm.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

Keywords

Elephant random walklaw of large numbersasymptotic (non)normalitymethod of momentsdifference equationMarkov chain

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Secondary: 60F15: Strong theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 805 - 829
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ben-Ari, I., Green, J., Meredith, T, Panzo, H. and Tan, X. (2019). Finite-memory elephant random walk and the central limit theorem for additive functionals. Available at arXiv:1911.05716.Google Scholar
Bercu, B. (2018). A martingale approach for the elephant random walk. J. Phys. A 81, 015201.CrossRefGoogle Scholar
Chen, A. and Renshaw, E.(1992). The Gillis–Domb–Fisher correlated random walk. J. Appl. Prob. 29, 792–813.CrossRefGoogle Scholar
Coletti, C. F., Gava, R. and Schütz, G. M. (2017). Central limit theorem and related results for the elephant random walk. J. Math. Phys. 58, 053303.CrossRefGoogle Scholar
Comets, F., Menshikov, M. V. and Wade, A. R. (2019). Random walks avoiding their convex hull with a finite memory. Available at arXiv:1902.09812.Google Scholar
Cressoni, J. C., da Silva, M. A. A. and Viswanathan, G. M. (2007). Amnestically induced persistence in random walks. J. Phys. A 46, 505002.CrossRefGoogle Scholar
Doob, J. L.(1953). Stochastic Processes. John Wiley, New York.Google Scholar
Engländer, J. and Volkov, S. (2018). Turning a coin over instead of tossing it. J. Theoret. Prob. 31, 1097–1118.CrossRefGoogle Scholar
González-Navarette, M. and Lambert, R. (2018). Non-Markovian random walks with memory lapses. J. Math. Phys. 59, 113301.CrossRefGoogle Scholar
Gut, A. (2013). Probability: A Graduate Course, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Herkenrath, U. (2003). A new approach to Markov processes of order 2. Ann. Univ. Craiova, Math. Comp. Sci. Ser. 30, 106–115.Google Scholar
Herkenrath, U., Iosifescu, M. and Rudolph, A. (2003). A note on invariance principles for iterated random functions. J. Appl. Prob. 40, 834–837.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters–Noordhof, Groningen.Google Scholar
Jones, G. L. (2004). On the Markov chain central limit theorem. Prob. Surveys 1, 299–320.CrossRefGoogle Scholar
Kelley, W. and Peterson, A. (2001). Difference Equations: An Introduction with Applications. Harcourt/Academic Press, San Diego.Google Scholar
Lyons, R. (1988). Strong laws of large numbers for weakly correlated random variables. Michigan Math. J. 35, 353–359.CrossRefGoogle Scholar
Menshikov, M. and Volkov, S. (2008). Urn-related random walk with drift $\rho x^{\alpha}/t^{\beta}$ . Electron. J. Prob. 13, 944–960.CrossRefGoogle Scholar
Moura, Th. R. S., Viswanathan, G. M., da Silva, M. A. A., Cressoni, J. C. and da Silva, L. R. (2016). Transient superdiffusion in random walks with a q-exponentially decaying memory profile. Physica A 453, 259–263.CrossRefGoogle Scholar
Schütz, G. M. and Trimper, S. (2004). Elephants can always remember: exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E, 70, 045101.CrossRefGoogle Scholar
da Silva, M. A. A., Cressoni, J. C. and Viswanathan, G. M. (2006). Discrete-time non-Markovian random walks: the effect of memory limitations on scaling. Physica A 364, 70–78.CrossRefGoogle Scholar