论文标题
多维大象随机步行的功能极限定理
Functional limit theorems for the Multi-dimensional Elephant Random Walk
论文作者
论文摘要
在本文中，我们将得出多维大象随机步行（MERW）的功能极限定理，从而扩展了Bercu and Laulin（2019）为一维边际提供的结果。 Merw是在$ \ Mathbb {z}^d $上进行的非马克维亚离散时间随机步行，它对整个过去都有完整的记忆，借此来寓言传统所说的大象永远不会忘记。顾名思义，Merw是大象随机步行（ERW）的$ D $维概括，后者首先是由Schütz和Trimper在2004年引入的。我们通过所谓的内存参数$ P $在零和一个之间来衡量大象记忆的影响。 Schütz和Trimper观察到的一个引人注目的功能是，ERW的长期行为在某些关键存储器参数$ P_C $上表现出相变。我们通过利用Merw和Pólyaurns之间的联系，遵循与Baur和Bertoin在ERW的工作中相似的思想，研究MER在所有内存制度中的渐近行为。
In this article we shall derive functional limit theorems for the multi-dimensional elephant random walk (MERW) and thus extend the results provided for the one-dimensional marginal by Bercu and Laulin (2019). The MERW is a non-Markovian discrete time-random walk on $\mathbb{Z}^d$ which has a complete memory of its whole past, in allusion to the traditional saying that an elephant never forgets. As the name suggests, the MERW is a $d$-dimensional generalisation of the elephant random walk (ERW), the latter was first introduced by Schütz and Trimper in 2004. We measure the influence of the elephant's memory by a so-called memory parameter $p$ between zero and one. A striking feature that has been observed by Schütz and Trimper is that the long-time behaviour of the ERW exhibits a phase transition at some critical memory parameter $p_c$. We investigate the asymptotic behaviour of the MERW in all memory regimes by exploiting a connection between the MERW and Pólya urns, following similar ideas as in the work by Baur and Bertoin for the ERW.
