The Beta(p,1) extensions of the random (uniform) Cantor sets

Dinis D. Pestana; Sandra M. Aleixo; J. Leonel Rocha

Discussiones Mathematicae Probability and Statistics (2009)

  • Volume: 29, Issue: 2, page 199-221
  • ISSN: 1509-9423

Abstract

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Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.

How to cite

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Dinis D. Pestana, Sandra M. Aleixo, and J. Leonel Rocha. "The Beta(p,1) extensions of the random (uniform) Cantor sets." Discussiones Mathematicae Probability and Statistics 29.2 (2009): 199-221. <http://eudml.org/doc/277045>.

@article{DinisD2009,
abstract = {Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.},
author = {Dinis D. Pestana, Sandra M. Aleixo, J. Leonel Rocha},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {order statistics; uniform spacings; random middle third Cantor set; Beta spacings; Hausdorff dimension; random Cantor set; Beta distribution; fractal},
language = {eng},
number = {2},
pages = {199-221},
title = {The Beta(p,1) extensions of the random (uniform) Cantor sets},
url = {http://eudml.org/doc/277045},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Dinis D. Pestana
AU - Sandra M. Aleixo
AU - J. Leonel Rocha
TI - The Beta(p,1) extensions of the random (uniform) Cantor sets
JO - Discussiones Mathematicae Probability and Statistics
PY - 2009
VL - 29
IS - 2
SP - 199
EP - 221
AB - Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.
LA - eng
KW - order statistics; uniform spacings; random middle third Cantor set; Beta spacings; Hausdorff dimension; random Cantor set; Beta distribution; fractal
UR - http://eudml.org/doc/277045
ER -

References

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  1. [1] S.M. Aleixo, Analytical Methods in Probability and Probabilistic Methods in Analysis: Fractality Arising in Connection to Beta(p,q) Models, Population Dynamics and Hausdorff Dimension, Ph. D. thesis, Univ. Lisbon, (2008). 
  2. [2] S.M. Aleixo, J.L. Rocha and D.D. Pestana, Populational Growth Models in the Light of Symbolic Dynamics, Proceedings of the ITI 2008, 30th International Conference on Information Technology Interfaces, Cavtat/Dubrovnik, Croatia (2008), IEEE:10.1109/ITI.2008.4588428, 311-316. 
  3. [3] S.M. Aleixo, J.L. Rocha and D.D. Pestana, Populational Growth Models Proportional to Beta Densities with Allee Effect, Proceedings of the BVP 2008, International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine, Spain, American Institute of Physics 1124 (2009), 3-12. 
  4. [4] S.M. Aleixo, J.L. Rocha and D.D. Pestana, Probabilistic Methods in Dynamical Analysis: Populations Growths Associated to Models Beta(p,q) with Allee Effect, Proceedings of the DYNA08, International Conference on Dynamics and Applications 2008, Braga, Portugal (2008), in press. Zbl1316.92061
  5. [5] S.M. Aleixo, J.L. Rocha and D.D. Pestana, Dynamical Behaviour on the Parameter Space: New Populational Growth Models Proportional to Beta Densities, Proceedings of the ITI 2009, 31th International Conference on Information Technology Interfaces, Cavtat/Dubrovnik, Croatia (2009), IEEE:10.1109/ITI.2009.5196082, 213-218. 
  6. [6] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley and Sons, New York 1990. 
  7. [7] Y. Pesin and H. Weiss, On the Dimension of the Deterministic and Random Cantor-Like Sets, Math. Res. Lett. 1 (1994), 519-529. Zbl0834.28004
  8. [8] D.D. Pestana, S.M. Aleixo and J.L. Rocha, Hausdorff Dimension of the Random Middle Third Cantor Set, Proceedings of the ITI 2009, 31th International Conference o n Information Technology Interfaces, Cavtat/Dubrovnik, Croatia (2009), IEEE:10.1109/ITI.2009.5196094, 279-284. 
  9. [9] J.L. Rocha and J.S. Ramos, Weighted Kneading Theory of One-Dimensional Maps with a Hole, Int. J. Math. Math. Sci. 38 (2004), 2019-2038. Zbl1068.37025
  10. [10] M. Schroeder, Fractals, Chaos, Power Laws: Minutes from a Infinite Paradise, W.H. Freeman, New York 1991. Zbl0758.58001
  11. [11] P. Waliszewski and J. Konarski, A Mystery of the Gompertz Function, Fractals in Biology and Medicine IV (2005), 277-286. 

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