A Random Walk with Strong Memory
DOI:
https://doi.org/10.52575/2687-0959-2026-58-2-183-194Keywords:
Random Walks, Non-markovian ProcessesAbstract
In this paper we study a particular random walk model where the walker can randomly revert to its previous positions. We propose a general semi-analytical approach to the calculation of the characteristic functions of the process. It is shown that under the uniform distribution of lags equations for the moments of the first two orders can be solved explicitly and the process itself has the limiting standard normal distribution after normalization. If the lags have exponential distribution with a small base, only some asymptotical estimations are available for the moments. Under other studied lag distributions, which do not allow even for such simple asymptotical analysis, direct numerical simulations have been performed. The study shows that the lag distribution with heavy tails and divergent expectation does not necessarily imply that the dispersion has power-law asymptotics, which is typical of anomalous diffusion.
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References
Samorodnitsky G. Stochastic processes and long range dependence. Springer International; 2016. 415 p.
Beran J., Feng Y., Ghosh S, Kulik R. Long-memory processes. Springer-Verlag Berlin; 2013. 884 p.
Schütz GM., Trimper S. Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk. Physical Review E. 2004;70(4):045101.
Laulin L. About the elephant random walk. PhD thesis. Universite´ de Bordeaux; 2022. 154 p.
Metzler R., Jeon J-H., Cherstvy AG., Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Physical Chemistry & Chemical Physics., 2014;16(44):24128–24164.
Kürsten R. Comment on «Anomalous diffusion induced by enhancement of memory». arXiv:1503.03302. 2015. 3 p.
Kumar N., Harbola U., Lindenberg K. Memory-induced anomalous dynamics: Emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model. Physical Review E. 2010;82(2):021101.
Beyer WA., Schrandt RG., Ulam SM. Computer studies of some history-dependent random processes. Technical report. Los Alamos National Lab; 1969. 9 p.
Kac M. A history-dependent random sequence defined by Ulam. Technical report. Los Alamos National Lab; 1969. 5 p.
Zenyuk DA. Discrete dynamical systems with random delays. Preprinty IPM im. M.V. Keldysha. 2024;70. 35 p. (In Russ.)
Clifford P., Stirzaker D. Reverting processes. arXiv:1911.07269. 2019. 15 p.
Gorenflo R., Mainardi F., Moretti D., Pagnini G, Paradisi P. Discrete random walk models for space-time fractional diffusion. Chemical Physics. 2002;284(1–2):521–541.
Roman S. The umbral calculus. Academic Press; 1984. 193 p.
Flajolet P., Sedgewick R. Analytic combinatorics. Cambridge University Press; 2009. 810 p.
Robbins H. The asymptotic distribution of the sum of a random number of random variables. Bulletin of the American Mathematical Society. 1948;54(12):1151–1161.
Kuczma M. Functional equations in a single variable. PWN Warszawa; 1968. 383 p.
Elaydi S. An introduction to difference equations. Springer New York; 2005. 562 p.
Karp RM. Probabilistic recurrence relations. Journal of the ACM. 1994;41(6):1136–1150.
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