Almost sure convergence and second moments of geometric functionals of fractal percolation

TL;DR

This paper investigates the almost sure limits and second moments of geometric measures of fractal percolation sets, extending previous work by analyzing convergence and covariance structures across all dimensions.

Contribution

It provides a comprehensive analysis of the almost sure limits, expectations, and covariances of intrinsic volumes of fractal percolation, including convergence rates, for any dimension.

Findings

Almost sure limits of rescaled intrinsic volumes are determined.

A factorization of limit variables allows explicit expectation and covariance calculations.

Rescaled expectations and variances converge to their limit counterparts with specified rates.

Abstract

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.