A martingale approach for the elephant random walk
By Bernard Bercu. 2017
135 citations recorded by the source corpus.
The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter p which lies between zero and one. This behavior is totally different in the diffusive regime 0⩽p<3/4, the critical regime p=3/4, and the superdiffusive regime 3/4<p⩽1. In the diffusive and critical regimes, we establish some new results on the almost sure asymptotic behavior of the ERW, such as the quadratic strong law and the law of the iterated logarithm. In the superdiffusive regime, we provide the first rigorous mathematical proof that the limiting distribution of the ERW is not Gaussian.
Subfields: Mathematical Physics, Molecular Biology, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/852558bd55f9f4eab8e74f9a0f5b84e8e9159c27
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Citations in the corpus
- On the Multi-dimensional Elephant Random Walk (2017, background): “egime p = pd. Theorem 3.6. We have the asymptotic normality (3.10) 1 √ nlogn Sn L −→ N 0, 1 d Id . 6 BERNARDBERCUANDLUCILELAULIN Remark 3.2. As before, in the special case d = 1, we find again [1],[3],[6] the asymptotic normality for the one-dimensional ERW √ Sn nlogn L −→ N(0,1). 3.3. The superdiff”