On the Multi-dimensional Elephant Random Walk
| Autor: | Lucile Laulin, Bernard Bercu |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Strong law
Probability (math.PR) Asymptotic distribution Statistical and Nonlinear Physics Law of the iterated logarithm Mathematics - Statistics Theory Statistics Theory (math.ST) Random walk 01 natural sciences 010305 fluids & plasmas [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Quadratic equation Convergence of random variables [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] 0103 physical sciences Multi dimensional Range (statistics) FOS: Mathematics Statistical physics 010306 general physics Mathematical Physics Mathematics - Probability ComputingMilieux_MISCELLANEOUS [PHYS.PHYS.PHYS-DATA-AN]Physics [physics]/Physics [physics]/Data Analysis Statistics and Probability [physics.data-an] Mathematics |
| Zdroj: | Journal of Statistical Physics Journal of Statistical Physics, Springer Verlag, 2019, 175 (6), pp.1146-1163. ⟨10.1007/s10955-019-02282-8⟩ |
| ISSN: | 0022-4715 1572-9613 |
| DOI: | 10.1007/s10955-019-02282-8⟩ |
| Popis: | The purpose of this paper is to investigate the asymptotic behavior of the multi-dimensional elephant random walk (MERW). It is a non-Markovian random walk which has a complete memory of its entire history. A wide range of literature is available on the one-dimensional ERW. Surprisingly, no references are available on the MERW. The goal of this paper is to fill the gap by extending the results on the one-dimensional ERW to the MERW. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the MERW. The asymptotic normality of the MERW, properly normalized, is also provided. In the superdiffusive regime, we prove the almost sure convergence as well as the mean square convergence of the MERW. All our analysis relies on asymptotic results for multi-dimensional martingales. |
| Databáze: | OpenAIRE |
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