论文标题
随机步行范围的能力：迭代对数定律
Capacity of the range of random walk: The law of the iterated logarithm
论文作者
论文摘要
我们同时建立了$ \ limsup $和迭代对数（lil）的$ \ liminf $ Law，以在任何维度$ d \ ge 3 $中进行简单随机步行的范围。虽然对于$ d \ ge 4 $，但在dimens $ d $匹配的$ n $ $ n $中的增长顺序与dimension $ d-2 $的随机步行范围的数量相匹配，有些令人惊讶的是，这种通信破坏了$ d = 3 $的范围的容量。我们进一步建立了这种LIL的Brownian容量，即$ 3 $维的Brownian样本路径和新颖的，急剧的中度偏差界限，可容纳$ 4 $维的简单随机步行的容量。
We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk.
