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Invariant Galton–Watson trees: metric properties and attraction with respect to generalized dynamical pruning

Part of: Stochastic processes Markov processes Partial differential equations

Published online by Cambridge University Press:  12 January 2023

Yevgeniy Kovchegov Open the ORCID record for Yevgeniy Kovchegov [Opens in a new window] ,
Guochen Xu  and
Ilya Zaliapin
Yevgeniy Kovchegov*
Affiliation:
Oregon State University
Guochen Xu*
Affiliation:
Oregon State University
Ilya Zaliapin*
Affiliation:
University of Nevada Reno
*
*Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA.
*Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA.
****Postal address: Department of Mathematics and Statistics, University of Nevada Reno, Reno, NV 89557-0172, USA. Email address: zal@unr.edu
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Abstract

The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

Keywords

Critical branching processrandom treeattractor

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G18: Self-similar processes 35B41: Attractors
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 643 - 671
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abraham, R. and Delmas, J. F. (2015). $\beta $ -coalescents and stable Galton–Watson trees. ALEA Lat. Am. J. Prob. Math. Statist. 12, 451–476.Google Scholar
Abramowitz, M. and Stegun, I. A. (eds) (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, DC.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Burd, G. A., Waymire, E. C. and Winn, R. D. (2000). A self-similar invariance of critical binary Galton–Watson trees. Bernoulli 6, 1–21.CrossRefGoogle Scholar
Devroye, L. and Kruszewski, P. (1994). A note on the Horton–Strahler number for random trees. Inf. Process. Lett. 56, 95–99.CrossRefGoogle Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes (Astérisque 281). Société Mathématique de France, Paris.Google Scholar
Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Prob. Theory Relat. Fields 139, 313–371.CrossRefGoogle Scholar
Duquesne, T. and Winkel, M. (2019). Hereditary tree growth and Lévy forests. Stoch. Process. Appl. 129, 3690–3747.CrossRefGoogle Scholar
Evans, S. N. (2008). Probability and Real Trees: École d’Été de Probabilités de Saint-Flour XXXV-2005. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Prob. Theory Relat. Fields 134, 81–126.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
He, H. and Winkel, M. (2014). Invariance principles for pruning processes of Galton–Watson trees. Preprint. Available at https://arxiv.org/abs/1409.1014.Google Scholar
Horton, R. E. (1945). Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology. Geol. Soc. Amer. Bull. 56, 275–370.CrossRefGoogle Scholar
Kesten, H. (1987) Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Prob. Statist. 22, 425–487.Google Scholar
Kovchegov, Y. and Zaliapin, I. (2016). Horton law in self-similar trees. Fractals 24, article no. 1650017.CrossRefGoogle Scholar
Kovchegov, Y. and Zaliapin, I. (2019). Random self-similar trees and a hierarchical branching process. Stoch. Process. Appl. 129, 2528–2560.CrossRefGoogle Scholar
Kovchegov, Y. and Zaliapin, I. (2020). Dynamical pruning of rooted trees with applications to 1D ballistic annihilation. J. Statist. Phys. 181, 618–672.CrossRefGoogle Scholar
Kovchegov, Y. and Zaliapin, I. (2020). Random self-similar trees: A mathematical theory of Horton laws. Prob. Surveys 17, 1–213.CrossRefGoogle Scholar
Kovchegov, Y. and Zaliapin, I. (2021). Invariance and attraction properties of Galton–Watson trees. Bernoulli 27, 1789–1823.CrossRefGoogle Scholar
Kovchegov, Y., Zaliapin, I. and Ben-Zion, Y. (2022) Invariant Galton–Watson branching process for earthquake occurrence. To appear in Geophys. J. Internat. Available at https://doi.org/10.1093/gji/ggac204.CrossRefGoogle Scholar
Le Jan, Y. (1991). Superprocesses and projective limits of branching Markov process. Ann. Inst. H. Poincaré Prob. Statist. 27, 91–106.Google Scholar
McConnell, M. and Gupta, V. (2008). A proof of the Horton law of stream numbers for the Tokunaga model of river networks. Fractals 16, 227–233.CrossRefGoogle Scholar
Neveu, J. (1986). Erasing a branching tree. Adv. Appl. Prob. 1, 101–108.Google Scholar
Neveu, J. and Pitman, J. (1989). Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. In Séminaire de Probabilités XXIII, Springer, Berlin, Heidelberg, pp. 239–247.CrossRefGoogle Scholar
Neveu, J. and Pitman, J. (1989). The branching process in a Brownian excursion. In Séminaire de Probabilités XXIII, Springer, Berlin, Heidelberg, pp. 248–257.CrossRefGoogle Scholar
Newman, W. I., Turcotte, D. L. and Gabrielov, A. M. (1997). Fractal trees with side branching. Fractals 5, 603–614.CrossRefGoogle Scholar
Peckham, S. D. (1995). New results for self-similar trees with applications to river networks. Water Resources Res. 31, 1023–1029.CrossRefGoogle Scholar
Strahler, A. N. (1957). Quantitative analysis of watershed geomorphology. Trans. Am. Geophys. Union 38, 913–920.CrossRefGoogle Scholar
Whittaker, E. T. and Watson, G. N. (1927). A Course of Modern Analysis, 4th edn. Cambridge University Press.Google Scholar
Zolotarev, V. M. (1957). More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 256–266.CrossRefGoogle Scholar