Local dimensions of the branching measure on a Galton–Watson tree

Quansheng Liu

Annales de l'I.H.P. Probabilités et statistiques (2001)

  • Volume: 37, Issue: 2, page 195-222
  • ISSN: 0246-0203

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Liu, Quansheng. "Local dimensions of the branching measure on a Galton–Watson tree." Annales de l'I.H.P. Probabilités et statistiques 37.2 (2001): 195-222. <http://eudml.org/doc/77687>.

@article{Liu2001,
author = {Liu, Quansheng},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Galton-Watson tree; branching measure; local dimension; exceptional points; uniform local dimensions; exact dimensions},
language = {eng},
number = {2},
pages = {195-222},
publisher = {Elsevier},
title = {Local dimensions of the branching measure on a Galton–Watson tree},
url = {http://eudml.org/doc/77687},
volume = {37},
year = {2001},
}

TY - JOUR
AU - Liu, Quansheng
TI - Local dimensions of the branching measure on a Galton–Watson tree
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2001
PB - Elsevier
VL - 37
IS - 2
SP - 195
EP - 222
LA - eng
KW - Galton-Watson tree; branching measure; local dimension; exceptional points; uniform local dimensions; exact dimensions
UR - http://eudml.org/doc/77687
ER -

References

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  1. [1] N.H Bingham, On the limit of a supercritical branching process, J. Appl. Prob.25A (1988) 215-228. Zbl0669.60078MR974583
  2. [2] N.H Bingham, R.A Doney, Asymptotic properties of supercritical branching processes I: The Galton–Watson process, Adv. Appl. Prob.6 (1974) 711-731. Zbl0297.60044MR362525
  3. [3] Y.S Chow, H Teicher, Probability: Independence, Interchageability and Martingales, Springer-Verlag, New York, 1978. Zbl0399.60001MR513230
  4. [4] C.S Dai, S.J Taylor, Defining fractals in a probability space, Ill. J. Math.38 (1994) 480-500. Zbl0797.60001MR1269700
  5. [5] A De Meyer, On a theorem of Bingham and Doney, J. Appl. Prob.19 (1982) 217-220. Zbl0481.60077MR644434
  6. [6] J.C D'Souza, The extinction time of the inhomogeneous branching Process, in: Heyde C.C (Ed.), Branching Processes: Proc. First World Congress, Lecture Notes in Statistics, 99, Springer, Berlin, 1995, pp. 106-117. Zbl0839.60078MR1351265
  7. [7] T.E Harris, Branching Processes, Springer-Verlag, 1963. MR163361
  8. [8] J Hawkes, A lower Lipschitz condition for the stable subordinator, Z. Wahr. verw. Geb.17 (1971) 23-32. Zbl0193.45002MR282413
  9. [9] J Hawkes, Trees generated by a simple branching process, J. London Math. Soc.24 (1981) 373-384. Zbl0468.60081MR631950
  10. [10] X Hu, S.J Taylor, The multifractal structure of stable occupation measure, Stoch. Proc. Appl.66 (1997) 283-299. Zbl0888.28004MR1440403
  11. [11] A Joffe, Remarks on the structure of trees with applications to supercritical Galton–Watson processes, in: Joffe A, Ney P (Eds.), Advances in Prob., 5, Dekker, New-York, 1978, pp. 263-268. Zbl0414.60078MR517537
  12. [12] P Lévy, Théorie de l'Addition des Variables Aléatoires, Gautier-Villars, Paris, 1954. Zbl0056.35903JFM63.0490.04
  13. [13] Q.S Liu, The exact Hausdorff dimension of a branching set, Prob. Theory Related Fieds104 (1996) 515-538. Zbl0842.60084MR1384044
  14. [14] Q.S Liu, The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale, Trees, in: Chauvin B, Cohen S, Rouault A (Eds.), Trees, Progress in Probability, 40, Birkhäuser, Basel, 1996, pp. 51-80. Zbl0864.60012MR1439972
  15. [15] Q.S Liu, Exact packing measure of the boundary of a Galton–Watson tree, Stoch. Proc. Appl.85 (2000) 19-28. Zbl0996.60094MR1730621
  16. [16] Q.S Liu, A Rouault, On two measures defined on the boundary of a branching tree, in: Athreya K.B, Jagers P (Eds.), Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, 84, Springer-Verlag, 1996, pp. 187-202. Zbl0867.60065MR1601741
  17. [17] Q.S Liu, N.R Shieh, A uniform limit law for the branching measure on a Galton–Watson tree, Asian J. Math.3 (1999) 381-386. Zbl0977.60078MR1796510
  18. [18] R Lyons, R Pemantle, Y Peres, Ergodic theory on Galton–Watson trees, Speed of random walk and dimension of harmonic measure, Ergodic Theory Dynamical Systems15 (1995) 593-619. Zbl0819.60077MR1336708
  19. [19] J Neveu, Arbre et processus de Galton–Watson, Ann. Inst. Henri Poincaré22 (1986) 199-207. Zbl0601.60082MR850756
  20. [20] G.L O'Brien, A limit theorem for sample maxima and heavy branches in Galton–Watson trees, J. Appl. Prob.17 (1980) 539-545. Zbl0428.60034MR568964
  21. [21] B Ramachandran, On the order and the type of entire characteristic functions, Ann. Stat.33 (1962) 1238-1255. Zbl0113.12804MR141145
  22. [22] N.R Shieh, S.J Taylor, Logarithmic multifractal spectrum of stable occupation measure, Stoch. Proc. Appl.75 (1998) 249-261. Zbl0932.60041MR1632209
  23. [23] N.R Shieh, S.J Taylor, Multifractal spectra of branching measure on a Galton–Watson tree, Preprint, 1999. Zbl1017.60043MR1895157

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