Probability Theory and Related Fields

, Volume 78, Issue 3, pp 335–355

Superbranching processes and projections of random Cantor sets

Authors

  • F. M. Dekking
    • Department of MathematicsDelft University of Technology
  • G. R. Grimmett
    • School of MathematicsUniversity of Bristol
Article

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

Abstract

We study sequences (X0, X1, ...) 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 Xn(m, i) are distributed as Xn and have certain properties of independence. We prove that, under appropriate conditions, Xn1/n→λ almost surely and in L1, where λ=sup E(Xn)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.

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© Springer-Verlag 1988