Abstract
We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We make here a slight abuse on the notation. Again, even though the dependency is not explicitly written, the discrete objects such as \(\kappa _n,\varvec{\mathbf {X}}^{(n)},\ldots \) all ultimately depend on the freezing threshold \(M\).
In the peeling of random Boltzmann maps [8], the locally largest cycles are called left-twigs.
We set here \(X_u(i):=0\) for \(i<0\) in order to not burden the notation with the indicator \(\mathbb {1}_{\{\beta _u\le k\}}\).
Strictly speaking, the results are only stated when there is no killing, that is \(\kappa ({q_*})=0\), but as mentioned by the authors [9, p. 2562, §2], they can be extended using the same techniques to the case where some killing is involved.
References
Aldous, D.: The continuum random tree. I. Ann. Probab. 19(1), 1–28 (1991)
Aldous, D.: The Continuum Random Tree. II. An Overview, Stochastic Analysis (Durham, 1990), London Mathematical Society Lecture Note series, vol. 167, pp. 23–70. Cambridge University Press, Cambridge (1991)
Aldous, D.: The continuum random tree. III. Ann. Probab. 21(1), 248–289 (1993)
Aspandiiarov, S., Iasnogorodski, R.: General criteria of integrability of functions of passage-times for non-negative stochastic processes and their applications. Teor. Veroyatnost. i Primenen. 43(3), 509–539 (1998)
Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York (1972). (Die Grundlehren der mathematischen Wissenschaften, Band 196)
Bertoin, J.: Markovian growth-fragmentation processes. Bernoulli 23(2), 1082–1101 (2017)
Bertoin, J., Budd, T., Curien, N., Kortchemski, I.: Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Relat. Fields 172(3–4), 663–724 (2018)
Bertoin, J., Curien, N., Kortchemski, I.: Random planar maps and growth-fragmentations. Ann. Probab. 46(1), 207–260 (2018)
Bertoin, J., Kortchemski, I.: Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab. 26(4), 2556–2595 (2016)
Bertoin, J., Stephenson, R.: Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21(66), 12 (2016)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1987)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin (2010). (corrected reprint of the second (1998) edition)
Evans, S.N.: Probability and real trees, lecture notes in mathematics. In: 2008, Lectures from the 35th Summer School on Probability Theory Held in Saint-Flour, July 6–23, vol. 1920, Springer, Berlin (2005)
Haas, B., Miermont, G.: The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9(4), 57–97 (2004)
Haas, B., Miermont, G.: Self-similar scaling limits of non-increasing Markov chains. Bernoulli 17(4), 1217–1247 (2011)
Haas, B., Miermont, G.: Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40(6), 2589–2666 (2012)
Haas, B., Miermont, G., Pitman, J., Winkel, M.: Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36(5), 1790–1837 (2008)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (1987)
Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)
Lamperti, J.: Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225 (1972)
Le Gall, J.-F.: Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15(1), 35–62 (2006)
Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)
Miermont, G.: Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Relat. Fields 131(3), 341–375 (2005)
Rembart, F., Winkel, M.: Recursive construction of continuum random trees. Ann. Probab. 46(5), 2715–2748 (2018)
Shi, Q.: Growth-fragmentation processes and bifurcators. Electron. J. Probab. 22, 25 (2017)
Shi, Z.: Branching random walks, lecture notes in mathematics. In: Lecture Notes from the 42nd Probability Summer School Held in Saint Flour, 2012, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School], vol. 2151, Springer, Cham (2015)
Acknowledgements
The author would like to thank Jean Bertoin for his constant suggestions and guidance and also Bastien Mallein for helpful discussions and comments. Thanks are also directed to an anonymous referee for their careful reading.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dadoun, B. Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes. J Theor Probab 33, 590–610 (2020). https://doi.org/10.1007/s10959-019-00975-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-019-00975-0