论文标题
几乎确定的收敛和分形渗透的几何功能的第二矩
Almost sure convergence and second moments of geometric functionals of fractal percolation
论文作者
论文摘要
我们确定几乎确定$ \ Mathbb {r}^d $的分形渗透步骤的固有量的限制，对于任何维度$ d \ geq 1 $。我们观察到这些极限变量的分解，特别是可以确定其期望和协方差结构。我们还显示了重新定位的期望和构建步骤内在体积的差异的融合到期望和限制变量的期望和差异，并在某些情况下给出了这种收敛的速率。这些结果大大扩展了我们先前的工作，该工作仅解决了内在体积期望的限制。
We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in $\mathbb{R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows, in particular, to determine their expectations and covariance structure. We also show convergence of rescaled expectations and variances of the intrinsic volumes of the construction steps to expectations and variances of the limit variables and give rates for this convergence in some cases. These results significantly extend our previous work that addressed only limits of expectations of intrinsic volumes.
