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Rates of Convergence in the Central Limit Theorem for the Elephant Random Walk with Random Step Sizes

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Abstract

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kolmogorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

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References

  1. Baur, E., Bertoin, J.: Elephant random walks and their connection to Pólya-type urns. Phys. Rev. E 94(5), 052134 (2016)

    Article  ADS  Google Scholar 

  2. Bercu, B.: A martingale approach for the elephant random walk. J. Phys. A: Math. Theor. 51, 015201 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bercu, B.: On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution. J. Stat. Phys. 189(1), 12 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bercu, B., Lucile, L.: On the multi-dimensional elephant random walk. J. Stat. Phys. 175(6), 1146–1163 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bercu, B., Lucile, L.: On the center of mass of the elephant random walk. Stoch. Process. Appl. 133(3), 111–128 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertoin, J.: Counting the zeros of an elephant random walk. Trans. Am. Math. Soc. 375, 5539–5560 (2022)

    MathSciNet  MATH  Google Scholar 

  7. Bikelis, A.: Estimates of the remainder term in the central limit theorem. Litovsk. Mat. Sb. 6, 323–346 (1966)

    MathSciNet  Google Scholar 

  8. Bobkov, S.G.: Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances. Probab. Theory Relat. Fields 170(1–2), 229–262 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bolthausen, E.: Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10, 672–688 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coletti, C.F., Gava, R., Schütz, G.M.: Central limit theorem and related results for the elephant random walk. J. Math. Phys. 58(5), 053303 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Coletti, C.F., Gava, R., Schütz, G.M.: A strong invariance principle for the elephant random walk. J. Stat. Mech. Theory Exp. 12, 123207 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dedecker, J., Merlevède, F., Rio, E.: Rates of convergence for minimal distances in the central limit theorem under projective criteria. Electron. J. Probab. 14, 978–1011 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dedecker, J., Merlevède, F., Rio, E.: Rates of convergence in the central limit theorem for martingales in the non stationary setting. Ann. Inst. H. Poincaré Probab. Statist. 58(2), 945–966 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dedecker, J., Merlevède, F., Rio, E.: Quadratic transportation cost in the conditional central limit theorem for dependent sequences. hal-03890107. To appear in Annales Henri Lebesgue (2022)

  15. Esseen, C.G.: On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astr. Fys. 28A(9), 19 (1942)

    MathSciNet  MATH  Google Scholar 

  16. Fan, X., Hu, H., Ma, X.: Cramér moderate deviations for the elephant random walk. J. Stat. Mech: Theory Exp. 2, 023402 (2021)

    Article  MATH  Google Scholar 

  17. Fan, X., Shao, Q.: Cramér’s moderate deviations for martingales with applications. Inst. Henri Poincaré Probab. Stat. Ann., to appear (2023)

  18. Grama, I., Haeusler, E.: Large deviations for martingales via Cramér’s method. Stoch. Process. Appl. 85, 279–293 (2000)

    Article  MATH  Google Scholar 

  19. Gut, A., Stadtmüller, U.: Variations of the elephant random walk. J. Appl. Probab. 58(3), 805–829 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Haeusler, E.: On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab. 16(1), 275–299 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. Academic, New York (1980)

    MATH  Google Scholar 

  22. Hayashi, M., Oshiro, S., Takei, M.: Rate of moment convergence in the central limit theorem for the elephant random walk. J. Stat. Mech.: Theory Exp. 2023, 023202 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu, Z., Feng, Q.: The enhanced strong invariance principle for the elephant random walk. Commun. Stat. Theory Methods (2022). https://doi.org/10.1080/03610926.2022.2092749

    Article  Google Scholar 

  24. Laulin, L.: New insights on the reinforced elephant random walk using a martingale approach. J. Stat. Phys. 186(1), 1–23 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Ma, X., El Machkouri, M., Fan, X.: On Wasserstein-1 distance in the central limit theorem for elephant random walk. J. Math. Phys. 63(1), 013301 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Rio, E.: Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45(3), 802–817 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Schütz, G.M., Trimper, S.: Elephants can always remember: exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E 70(4), 045101 (2004)

    Article  ADS  Google Scholar 

  28. Shao, Q.M.: Almost sure invariance principles for mixing sequences of random variables. Stoch. Process. Appl. 48(2), 319–334 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vázquez Guevara, V.H.: On the almost sure central limit theorem for the elephant random walk. J. Phys. A: Math. Theor. 52(47), 475201 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the referees for their suggestions and helpful comments. The work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11971063 and 12371155).

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Authors and Affiliations

  1. Université Paris Cité, Laboratoire MAP5 and CNRS UMR 8145, 75016, Paris, France

    Jérôme Dedecker

  2. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, China

    Xiequan Fan & Haijuan Hu

  3. LAMA, Univ Gustave Eiffel, Univ Paris Est Créteil, UMR 8050 CNRS, 77454, Marne-La-Vallée, France

    Florence Merlevède

Authors
  1. Jérôme Dedecker
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  2. Xiequan Fan
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  3. Haijuan Hu
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  4. Florence Merlevède
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Correspondence to Xiequan Fan.

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Communicated by Gregory Schehr.

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Dedecker, J., Fan, X., Hu, H. et al. Rates of Convergence in the Central Limit Theorem for the Elephant Random Walk with Random Step Sizes. J Stat Phys 190, 154 (2023). https://doi.org/10.1007/s10955-023-03168-6

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  • DOI: https://doi.org/10.1007/s10955-023-03168-6

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