点过程的滑动维数
The Slide Dimension of Point Processes
摘要：
我们将度量空间的任何有限子集与尺度不变数的无限序列$ρ_1,ρ_2,\dots$相关联，该序列是从称为类熵的微分熵的变体导出的。作为点过程的统计，这些数字往往出现在模拟收敛，我们给出的例子，其中$1/ρ_1$收敛到Hausdorff维。本文利用$ρ_n$定义了度量空间上一类特殊点过程的滑动维数。本文用滑动微积分定义了$ρ_n$，并导出了计算$ρ_1$的显式公式。对于$[0,1]^n$上的均匀随机变量X，给出了$ρ_1(X) =1/n$和$ρ_2(X) =-π^2/(6n^2)$的证据，并与正态变量$Z$的模拟表明，$ρ_1(Z) =4/π$和$ρ_2(Z) =-1$。一些潜在的应用空间统计被认为是。
英文摘要：
We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $ρ_1,ρ_2,\dots$ derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where $1/ρ_1$ converges to the Hausdorff dimension. We use the $ρ_n$ to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define $ρ_n$ and an explicit formula is derived for the calculation of $ρ_1$. For a uniform random variable X on $[0,1]^n$, evidence is given that $ρ_1(X) =1/n$ and $ρ_2(X) =-π^2/(6n^2)$ and simulations with a normal variable $Z$ suggest that $ρ_1(Z) =4/π$ and $ρ_2(Z) =-1$. Some potential applications to spatial statistics are considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
