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Superbranching processes and projections of random Cantor sets
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  • Published: July 1988

Superbranching processes and projections of random Cantor sets

  • F. M. Dekking1 &
  • G. R. Grimmett2 

Probability Theory and Related Fields volume 78, pages 335–355 (1988)Cite this article

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  • 30 Citations

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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 1/n 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Delft University of Technology, Julianalaan 132, 2628 BL, Delft, The Netherlands

    F. M. Dekking

  2. School of Mathematics, University of Bristol, University Walk, BS8 1TW, Bristol, United Kingdom

    G. R. Grimmett

Authors
  1. F. M. Dekking
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  2. G. R. Grimmett
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Additional information

Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant

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Cite this article

Dekking, F.M., Grimmett, G.R. Superbranching processes and projections of random Cantor sets. Probab. Th. Rel. Fields 78, 335–355 (1988). https://doi.org/10.1007/BF00334199

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  • Received: 08 July 1987

  • Revised: 02 February 1988

  • Issue Date: July 1988

  • DOI: https://doi.org/10.1007/BF00334199

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Keywords

  • Positive Integer
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Lebesgue Measure
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