arXiv 1302.29712013-02math.PR

平面对称简单随机游动的局部极限定理与旋转对称性的丧失

A Local Limit Theorem and Loss of Rotational Symmetry of Planar Symmetric Simple Random Walk

Christian Beneš

备注：: 20 pages, 3 figures
发表信息：: Trans. Amer. Math. Soc. 371 (2019), 2553-2573
DOI：: 10.1090/tran/7399

摘要：

我们推导出一个局部极限定理的正常，适度，和大偏差对称简单随机游动的正方形格子在尺寸1和2，是一个改进的现有结果的点，特别是远离步行的起点。更具体地说，我们给出了明确的渐近表达式$n$和$x$，其中$x$被认为是依赖于$n$，在一维和二维$P(S_n=x)$，对称简单随机游动$S$开始在原点的概率是在某个点$x$在时间$n$，这是有效的所有$x$。我们还表明，平面对称的简单随机游动的行为从根本上不同于平面标准布朗运动的半径$n^{3/4}$，其中的磁盘以外的随机游动不再是近似旋转对称的。实际上，如果是$n^{3/4}=o(|S_n|)$，则更有可能沿着坐标轴找到$S_n$。本文描述了$S$从近似旋转对称到完全集中的转变过程。

英文摘要：

We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S_n|)$, $S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.

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