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.
Article PDF
Similar content being viewed by others
References
Athreya, K.B., Karlin, S.: Branching processes with random environments I: extinction probabilities. Ann. Math. Statist. 42, 1499–1520 (1971)
Athreya, K.B., Ney, P.E.: Branching processes. Berlin Heidelberg New York: Springer 1972
Biggins, J.D.: Chernoff's theorem in the branching random walk, J. Appl. Probab. 14, 630–636 (1977)
Biggins, J.D.: Growth rates in the branching random walk, Z. Wahrscheinlichkeitstheor. Verw. Geb. 48, 17–34 (1979)
Dekking, F.M.: Subcritical branching processes in a two state random environment, and a percolation probelm on trees. J. Appl. Probab., 24, 798–808 (1987)
Dekking, F.M.: On the survival probability of a branching process in a finite state i.i.d. environment, Stoch. Proc. Appl. in press
Falconer, K.J.: The geometry of fractal sets. Cambridge: Cambridge University Press 1985
Falconer, K.J.: Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582 (1986)
Falconer, K.J.: Cut-set sums and tree processes, Proc. Am. Math. Soc. 101, 337–346 (1987)
Graf, S.: Statistically self-similar fractals. Probab. Th. Rel. Fields 74, 357–392 (1987)
Hawkes, J.: Hausdorff measure, entropy, and the independence of small sets, Proc. London Math. Soc. (3), 28, 700–724 (1974)
Hawkes, J.: Trees generated by a simple branching process. J. London Math. Soc. (2), 24, 373–384 (1981)
Mandelbrot, B.: The fractal geometry of nature, New York: Freeman 1983
Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346 (1986)
Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3), 4, 257–302 (1954)
Neveu, J.: Arbres et processus de Galton-Watson, Ann. Inst. Henri Poincaré 22, 199–207 (1986)
Peyrière, J.: Mandelbrot random beadsets and birthprocesses with interaction, I.B.M. research report RC-7417 1978
Tricot, C.: Douze définitions de la densité logarithmique. C.R. Acad. Sc. Paric 293, 549–552 (1981)
Author information
Authors and Affiliations
Additional information
Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant
Rights and permissions
About this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00334199