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 quadratric strong law for the center of mass of the elephant random walk. The asymptotic normality of the center of mass, 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.

中文翻译：

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关于大象随机游走的质心

我们的目标是研究大象随机游走的质心的渐近行为，这是对整数的离散时间随机游走，具有对其整个历史的完整记忆。在扩散和临界状态下，我们为大象随机游走的质心建立了几乎肯定收敛、迭代对数定律和二次强定律。还提供了经过适当归一化的质心的渐近正态性。最后，我们证明了超扩散区域质心的强极限定理。我们所有的分析都依赖于多维鞅的渐近结果。