论文标题
随机行走高维极限I：Wiener螺旋
Random Walks in the High-Dimensional Limit I: The Wiener Spiral
论文作者
论文摘要
我们证明，在$ d $ d $二维的欧几里得空间中，$ n $ steps的随机步行限制了定理，因为$ n $ and $ n $ and $ d $倾向于无限。我们的结果之一指出，这种随机步行的路径被视为无限维二维希尔伯特空间的紧凑子集$ \ ell^2 $，在Hausdorff距离距离距离距离距离，并在gromov-hausdorff中，以及在gromov-hausdorff中，也向维也纳的螺旋形上，如$ d，n \ n \ for。另一组结果描述了在时间$ n $和原点之间随机步行者之间的平方距离的各种可能的限制分布。
We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the infinite-dimensional Hilbert space $\ell^2$, converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense to the Wiener spiral, as $d,n\to\infty$. Another group of results describes various possible limit distributions for the squared distance between the random walker at time $n$ and the origin.
