## Hausdorff dimension of limsup random fractals.(English)Zbl 1287.60018

A formula for the Hausdorff dimension of limsup random fractals is derived. Consider the collection $$\mathcal{D}_n$$ of all $$N$$-dimensional hyper-cubes having the form $$[k^12^{-n}, (k^1+1)2^{-n}] \times \cdots \times [k^N2^{-n}, (k^N+1)2^{-n}]$$ for an $$N$$-dimensional positive integer $$k \in \mathbb{Z}_+^N$$. Let $$\{ Z_n(I):I \in \mathcal{D}_n\}$$ be a collection of Bernoulli r.v.’s such that $$\Pr \left\{ Z_n=1 \right\} = q_n$$. A limsup random fractal is $$A:= \bigcap_{n \geq 1} \bigcup_{k \geq n} A_k$$, where $$A_k= \bigcup_{I \in \mathcal{D}_k, Z_k (I) = 1} I^\circ$$ and $$I^\circ$$ is the interior of $$I$$.
Under the assumption that the Bernoulli random variables are independent, the author derives the following formula for the Hausdorff dimension of a limsup random fractals defined on the boundary of a spherically symmetric tree ($$T$$) $\left\| \dim_H(A) \right\|_{L^\infty \left( P \right)} \, = \, \sup \left\{ s \geq 0 : \delta_s(\partial T) > t \right\}$ for $\delta_s(\partial T) := \lim_{n \to \infty} \frac{1}{-n} \log \left( \inf_{\mu \in \mathcal{P}(\partial T)} \iint_{d(x, y) \leq e^{-n}} d(x, y)^{-s} \mu(dx) \mu(dy) \right),$ where $$d$$ is the metric of the tree on the boundary and $$\mathcal{P}(\partial T)$$ is the collection of all Borel probability measures supported on $$\partial T$$.

### MSC:

 60D05 Geometric probability and stochastic geometry 28A80 Fractals
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