We introduce a new random walk with unbounded memory obtained as a mixture of the Elephant Random Walk and the Dynamic Random Walk which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold.

中文翻译：

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内存受动态系统扰动的随机游走的极限定理

我们引入了一种具有无限内存的新随机游走，它是 Elephant Random Walk 和 Dynamic Random Walk 的混合体，我们称之为 Dynamic Elephant Random Walk (DERW)。由于这种混合，所产生的随机过程的增量分布与时间有关。我们证明了 DERW 的强大的大数定律，并且在特定情况下，我们为其速度提供了明确的表达式。最后，我们给出了中心极限定理和迭代对数定律成立的充分条件。