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Introducing smooth amnesia to the memory of the elephant random walk. (English) Zbl 1500.60019

Summary: This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

MSC:

60F17 Functional limit theorems; invariance principles
60G42 Martingales with discrete parameter
60G50 Sums of independent random variables; random walks
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