The Annals of Probability

Exact Hausdorff measure on the boundary of a Galton–Watson tree

Toshiro Watanabe

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A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton–Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions ϕ according to whether ϕ-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes’s conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.

Article information

Ann. Probab., Volume 35, Number 3 (2007), 1007-1038.

First available in Project Euclid: 10 May 2007

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A78: Hausdorff and packing measures
Secondary: 60G18: Self-similar processes 28A80: Fractals [See also 37Fxx]

Galton–Watson tree exact Hausdorff measure shift self-similar additive random sequence boundary branching measure dominated variation b-decomposable distribution


Watanabe, Toshiro. Exact Hausdorff measure on the boundary of a Galton–Watson tree. Ann. Probab. 35 (2007), no. 3, 1007--1038. doi:10.1214/009117906000000629.

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  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Bass, R. F. and Kumagai, T. (2000). Laws of the iterated logarithm for some symmetric diffusion processes. Osaka J. Math. 37 625--650.
  • Berlinkov, A. (2003). Exact packing dimension in random recursive constructions. Probab. Theory Related Fields 126 477--496.
  • Berlinkov, A. and Mauldin, R. D. (2002). Packing measure and dimension of random fractals. J. Theoret. Probab. 15 695--713.
  • Biggins, J. D. and Bingham, N. H. (1991). Near-constancy phenomena in branching processes. Math. Proc. Cambridge Philos. Soc. 110 545--558.
  • Biggins, J. D. and Bingham, N. H. (1993). Large deviations in the supercritical branching process. Adv. in Appl. Probab. 25 757--772.
  • Biggins, J. D. and Nadarajah, S. (1993). Near-constancy of the Harris function in the simple branching process. Comm. Statist. Stochastic Models 9 435--444.
  • Bingham, N. H. (1988). On the limit of a supercritical branching process. J. Appl. Probab. 25 215--228.
  • Bingham, N. H. and Doney, R. A. (1974). Asymptotic properties of supercritical branching processes. I. The Galton--Watson process. Adv. in Appl. Probab. 6 711--731.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Bunge, J. (1997). Nested classes of $C$-decomposable laws. Ann. Probab. 25 215--229.
  • Falconer, K. J. (1986). Random fractals. Math. Proc. Cambridge Philos. Soc. 100 559--582.
  • Fukushima, M., Shima, T. and Takeda, M. (1999). Large deviations and related LIL's for Brownian motions on nested fractals. Osaka J. Math. 36. 497--537.
  • Graf, S., Mauldin, R. D. and Williams, S. C. (1988). The Exact Hausdorff Dimension in Random Recursive Constructions. Amer. Math. Soc., Providence, RI.
  • Hambly, B. M. and Jones, O. D. (2003). Thick and thin points for random recursive fractals. Adv. in Appl. Probab. 35 251--277.
  • Harris, T. E. (1948). Branching processes. Ann. Math. Statist. 19 474--494.
  • Hawkes, J. (1981). Trees generated by a simple branching process. J. London Math. Soc. (2) 24 373--384.
  • Holmes, R. A. (1973). Local asymptotic law and the exact Hausdorff measure for a simple branching process. Proc. London Math. Soc. (3) 26 577--604.
  • Kasahara, Y. (1978). Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18 209--219.
  • Lalley, S. P. and Sellke, T. (2000). An extension of Hawkes' theorem on the Hausdorff dimension of a Galton--Watson tree. Probab. Theory Related Fields 116 41--56.
  • Liu, Q. S. (1996). The exact Hausdorff dimension of a branching set. Probab. Theory Related Fields 104 515--538.
  • Liu, Q. S. (2000). Exact packing measure on a Galton--Watson tree. Stochastic. Process. Appl. 85 19--28.
  • Liu, Q. S. (2001). Local dimension of the branching measure on a Galton--Watson tree. Ann. Inst. H. Poincaré Probab. Statist. 37 195--222.
  • Liu, Q. S. and Shieh, N.-R. (1999). A uniform limit law for the branching measure on a Galton--Watson tree. Asian J. Math. 3 381--386.
  • Loève, M. (1945). Nouvelles classes de loi limites. Bull. Soc. Math. France 73 107--126.
  • Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931--958.
  • Maejima, M. and Naito, Y. (1998). Semi-selfdecomposable distributions and a new class of limit theorems. Probab. Theory Related Fields 112 13--31.
  • Maejima, M. and Sato, K. (1999). Semi-selfsimilar processes. J. Theoret. Probab. 12 347--373.
  • Maejima, M., Sato, K. and Watanabe, T. (1999). Operator semi-selfdecomposability, ($C,Q$)-decomposability and related nested classes. Tokyo J. Math. 22 473--509.
  • Maejima, M., Sato, K. and Watanabe, T. (2000). Completely operator semi-selfdecomposable distributions. Tokyo J. Math. 23 235--253.
  • Maejima, M., Sato, K. and Watanabe, T. (2000). Distributions of selfsimilar and semi-selfsimilar processes with independent increments. Statist. Probab. Lett. 47 395--401.
  • de Meyer, A. (1982). On a theorem of Bingham and Doney. J. Appl. Probab. 19 217--220.
  • Mörters, P. and Shieh, N.-R. (2002). Thin and thick points for branching measure on a Galton--Watson tree. Statist. Probab. Lett. 58 13--22.
  • Mörters, P. and Shieh, N. R. (2004). On the multifractal spectrum of the branching measure of a Galton--Watson tree. J. Appl. Probab. 41 1223--1229.
  • Pruitt, W. E. (1990). The rate of escape of random walk. Ann. Probab. 18 1417--1461.
  • Sato, K. (1991). Self-similar processes with independent increments. Probab. Theory Related Fields 89 285--300.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Shieh, N. R. and Taylor, S. J. (2002). Multifractal spectra of branching measure on a Galton--Watson tree. J. Appl. Probab. 39 100--111.
  • Shimura, T. and Watanabe, T. (2005). Infinite divisibility and generalized subexponentiality. Bernoulli 11 445--469.
  • Taylor, S. J. (1986). The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 383--406.
  • Watanabe, T. (1996). Sample function behavior of increasing processes of class $L$. Probab. Theory Related Fields 104 349--374.
  • Watanabe, T. (2000). Continuity properties of distributions with some decomposability. J. Theoret. Probab. 13 169--191.
  • Watanabe, T. (2000). Absolute continuity of some semi-selfdecomposable distributions and self-similar measures. Probab. Theory Related Fields 117 387--405.
  • Watanabe, T. (2002). Limit theorems for shift selfsimilar additive random sequences. Osaka J. Math. 39 561--603.
  • Watanabe, T. (2002). Shift self-similar additive random sequences associated with supercritical branching processes. J. Theoret. Probab. 15 631--665.
  • Watanabe, T. (2004). Exact packing measure on the boundary of a Galton--Watson tree. J. London Math. Soc. (2) 69 801--816.
  • Wolfe, S. J. (1983). Continuity properties of decomposable probability measures on Euclidean spaces. J. Multivariate Anal. 13 534--538.