On the center of mass of the elephant random walk. (English) Zbl 1469.60135
Summary: Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the center of mass of the elephant random walk. The asymptotic normality, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60G42 | Martingales with discrete parameter |
| 60F05 | Central limit and other weak theorems |
Keywords:
elephant random walk; center of mass; multi-dimensional martingales; almost sure convergence; asymptotic normalityReferences:
| [1] | Baur, E.; Bertoin, J., Elephant random walks and their connection to Pólya-type urns, Phys. Rev. E, 94, Article 052134 pp. (2016) |
| [2] | Bercu, B., A martingale approach for the elephant random walk, J. Phys. A, 51, 1, Article 015201 pp. (2018), 16 · Zbl 1392.60038 |
| [3] | Bercu, B.; Chabanol, M.-L.; Ruch, J.-J., Hypergeometric identities arising from the elephant random walk, J. Math. Anal. Appl., 480, 1, Article 123360 pp. (2019), 12 · Zbl 1479.33003 |
| [4] | Bercu, B.; Laulin, L., On the multi-dimensional elephant random walk, J. Stat. Phys., 175, 6, 1146-1163 (2019) · Zbl 1421.62116 |
| [5] | Bertenghi, M., Functional limit theorems for the multi-dimensional elephant random walk (2020), arXiv:2004.02004 |
| [6] | Bertoin, J., Scaling exponents of step-reinforced random walks (2020), hal-02480479 |
| [7] | Businger, S., The shark random swim (Lévy flight with memory), J. Stat. Phys., 172, 3, 701-717 (2018) · Zbl 1400.82097 |
| [8] | Chaabane, F.; Maaouia, F., Théorèmes limites avec poids pour les martingales vectorielles, ESAIM Probab. Statist., 4, 137-189 (2000) · Zbl 0965.60033 |
| [9] | Coletti, C. F.; Gava, R.; Schütz, G. M., Central limit theorem and related results for the elephant random walk, J. Math. Phys., 58, 5, Article 053303 pp. (2017), 8 · Zbl 1375.60086 |
| [10] | Coletti, C. F.; Gava, R.; Schütz, G. M., A strong invariance principle for the elephant random walk, J. Stat. Mech. Theory Exp., 12, Article 123207 pp. (2017), 8 · Zbl 1457.82148 |
| [11] | Coletti, C.; Papageorgiou, I., Asymptotic analysis of the elephant random walk (2019), arXiv:1910.03142 |
| [12] | Duflo, M., Random iterative models, (Applications of Mathematics (New York), vol. 34 (1997), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0868.62069 |
| [13] | Fan, X.; Hu, H.; Xiaohui, M., Some limit theorems for the elephant random walk (2020), arXiv:2003.12937 |
| [14] | Grill, K., On the average of a random walk, Statist. Probab. Lett., 6, 5, 357-361 (1988) · Zbl 0686.60068 |
| [15] | Hall, P.; Heyde, C. C., Martingale limit theory and its application, Probability and Mathematical Statistics (1980), Academic Press, Inc.: Academic Press, Inc. New York-London · Zbl 0462.60045 |
| [16] | Kabluchko, Z.; Vysotsky, V.; Zaporozhets, D., Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, Geom. Funct. Anal., 27, 4, 880-918 (2017) · Zbl 1373.52009 |
| [17] | Kubota, N.; Takei, M., Gaussian fluctuation for superdiffusive elephant random walks, J. Stat. Phys., 177, 6, 1157-1171 (2019) · Zbl 1439.60043 |
| [18] | Lo, C. H.; Wade, A. R., On the centre of mass of a random walk, Stochastic Process. Appl., 129, 11, 4663-4686 (2019) · Zbl 1448.60097 |
| [19] | McRedmond, J.; Wade, A. R., The convex hull of a planar random walk: perimeter, diameter, and shape, Electron. J. Probab., 23 (2018), Paper No. 131, 24 · Zbl 1406.60068 |
| [20] | Miyazaki, T.; Takei, M., Limit theorems for the ‘laziest’ minimal random walk model of elephant type, J. Stat. Phys., 181, 2, 587-602 (2020) · Zbl 1460.60036 |
| [21] | Schütz, G. M.; Trimper, S., Elephants can always remember: Exact long-range memory effects in a non-markovian random walk, Phys. Rev. E, 70, Article 045101 pp. (2004) |
| [22] | Stout, W. F., Maximal inequalities and the law of the iterated logarithm, Ann. Probab., 1, 322-328 (1973) · Zbl 0262.60016 |
| [23] | Touati, A., Sur la convergence en loi fonctionnelle de suites de semimartingales vers un mélange de mouvements browniens, Teor. Veroyatn. Primen., 36, 4, 744-763 (1991) · Zbl 0762.60026 |
| [24] | V., Strassen., An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 3, 1964, 211-226 (1964) · Zbl 0132.12903 |
| [25] | Vázquez Guevara, V. H., On the almost sure central limit theorem for the elephant random walk, J. Phys. A, 52, 1, Article 475201 pp. (2019) · Zbl 1509.82063 |
| [26] | Vysotsky, V.; Zaporozhets, D., Convex hulls of multidimensional random walks, Trans. Amer. Math. Soc., 370, 11, 7985-8012 (2018) · Zbl 1434.60060 |
| [27] | Wade, A. R.; Xu, C., Convex hulls of planar random walks with drift, Proc. Amer. Math. Soc., 143, 1, 433-445 (2015) · Zbl 1311.60052 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
