# Superbranching processes and projections of random Cantor sets

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DOI: 10.1007/BF00334199

- Cite this article as:
- Dekking, F.M. & Grimmett, G.R. Probab. Th. Rel. Fields (1988) 78: 335. doi:10.1007/BF00334199

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## Abstract

We study sequences (*X*_{0}, *X*_{1}, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that \(X_{m + n} \geqq \sum\limits_{i = 1}^{X_m } {X_n (m,i)}\), for *m*, *n*≧0, where the *X*_{n}*(m, i)* are distributed as *X*_{n} and have certain properties of independence. We prove that, under appropriate conditions, *X*_{n}^{1/n}→λ almost surely and in *L*^{1}, where λ=sup *E(X*_{n})^{1/n}. Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the *M*-adic subcubes of [0, 1]^{d}. We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.