Geodesic
Galton–Watson processes and their role as building blocks for branching processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 1, pp. 177-192 Cet article a éte moissonné depuis la source Math-Net.Ru

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This article is an essay, both expository and argumentative, on the Galton–Watson process as a tool in the domain of branching processes. It is at the same time the author's way of honoring two distinguished scientists in this domain, both from the Russian Academy of Sciences, and congratulating them on their special birthdays. The thread of the article is the role which the Galton–Watson process had played in the author's own research. We start with a discussion on a controlled Galton–Watson process. Then we pass to random absorbing processes, and also recall and discuss a problem in medicine. Further questions bring us, via the Borel–Cantelli lemma, to $\varphi$-branching processes and extensions. To gain more generality we then look at bisexual Galton–Watson processes. Finally, we briefly discuss relatively complicated resource-dependent branching processes to show that, again, using Galton–Watson reproduction schemes (whenever reasonable) can be a convincing approach to new processes, which are then sufficiently tractable to obtain results of interest.
Keywords: controlled branching process, $\varphi$-branching process, bisexual reproduction, Borel–Cantelli lemma, resource dependence, society forms, stopping times, theorem of envelopment, BRS-inequality. 
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F. Thomas Bruss. Galton–Watson processes and their role as building blocks for branching processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 1, pp. 177-192. http://geodesic.mathdoc.fr/item/TVP_2022_67_1_a10/

[1] V. I. Afanasyev, J. Geiger, G. Kersting, V. A. Vatutin, “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673 | DOI | MR | Zbl

[2] O. Barndorff-Nielsen, On the rate of growth of partial extrema of independent, identically distributed random variables, Amanuensis of Mathematics, Techn. Scient. Note 5, 1961, 17 pp.

[3] D. Bertacchi, F. P. Machado, F. Zucca, “Local and global survival for nonhomogeneous random walk systems on $\mathbb Z$”, Adv. in Appl. Probab., 46 (2014), 256–278 | DOI | MR | Zbl

[4] C. Biró, I. R. Curbelo, “Weak independence of events and the converse of the Borel–Cantelli Lemma”, Expo. Math., 2021, 1–13, Publ. online ; 2021 (v1 – 2020), 12 pp. | DOI

[5] F. T. Bruss, “Branching processes with random absorbing processes”, J. Appl. Probab., 15:1 (1978), 54–64 | DOI | MR | Zbl

[6] F. T. Bruss, “How to apply a medicament, if its direct efficiency is unknown”, Ann. Soc. Sci. Bruxelles Sér. I, 93:1 (1979), 39–54 | Zbl

[7] F. T. Bruss, “A counterpart of the Borel–Cantelli Lemma”, J. Appl. Probab., 17:4 (1980), 1094–1101 | DOI | MR | Zbl

[8] F. T. Bruss, “A note on extinction criteria for bisexual Galton–Watson processes”, J. Appl. Probab., 21:4 (1984), 915–919 | DOI | MR | Zbl

[9] F. T. Bruss, “Resource dependending branching processes”, in “Eleventh conference on stochastic processes and their applications: Clermont-Ferrand, France, 28 June–2 July 1982”, Stochastic Process. Appl., 16:1 (1984), 36 | DOI | Zbl

[10] F. T. Bruss, M. Slavtchova-Bojkova, “On waiting times to populate an environment and a question of statistical inference”, J. Appl. Probab., 36:1 (1999), 261–267 | DOI | MR | Zbl

[11] F. T. Bruss, M. Duerinckx, “Resource dependent branching processes and the envelope of societies”, Ann. Appl. Probab., 25:1 (2015), 324–372 | DOI | MR | Zbl

[12] F. T. Bruss, “The BRS-inequality and its applications”, Probab. Surv., 18 (2021), 44–76 | DOI | MR | Zbl

[13] S. N. Cohen, V. Fedyashov, Ergodic BSDEs with jumps and time dependence, 2015 (v1 – 2014), 37 pp., arXiv: 1406.4329v2

[14] D. J. Daley, “Extinction conditions for certain bisexual Galton–Watson branching processes”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9:4 (1968), 315–322 | DOI | MR | Zbl

[15] K. Dietz, “Simulation models for genetic control alternatives”, Mathematical analysis of decision problems in ecology (Istanbul, 1973), Lecture Notes in Biomath., 5, Springer-Verlag, Berlin–New York, 1975, 299–317 | DOI

[16] N. D. Feldheim, O. N. Feldheim, “Convergence of the quantile admission process with veto power”, Stochastic Process. Appl., 130:7 (2020), 4294–4325 | DOI | MR | Zbl

[17] W. M. Feldman, P. E. Souganidis, “Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations”, J. Math. Pures Appl. (9), 108:5 (2017), 751–782 | DOI | MR | Zbl

[18] M. González Velasco, I. M. del Puerto García, G. P. Yanev, Controlled branching processes, v. 2, ed. E. Yarovaya, ISTE Ltd., London; John Wiley Sons, Inc., Hoboken, NJ, 2017, 240 pp.

[19] M. Gonzáles, C. Minuesa, I. del Puerto, “Maximum likelihood estimation and expectation-maximization algorithm for controlled branching processes”, Comput. Statist. Data Anal., 93 (2016), 209–227 | DOI | MR | Zbl

[20] P. Haccou, P. Jagers, V. A. Vatutin, Branching processes. Variation, growth, and extinction of populations, Camb. Stud. Adapt. Dyn., 5, Cambridge Univ. Press, Cambridge, 2005, xii+316 pp. | DOI | MR | Zbl

[21] G. K. Kobanenko, “Limit theorems for bounded branching processes”, Discrete Math. Appl., 28:5 (2018), 285–292 | DOI | DOI | MR | Zbl

[22] A. Makur, E. Mossel, Y. Polyanskiy, Broadcasting on two-dimensional regular grids, 2020, 52 pp., arXiv: 2010.01390v1

[23] M. Molina, M. Mota, A. Ramos, “Statistical inference in two-sex biological populations with reproduction in a random environment”, Ecol. Complex., 30 (2017), 63–69 | DOI

[24] H.-J. Schuh, “A condition for the extinction of a branching process with an absorbing lower barrier”, J. Math. Biol., 3 (1976), 271–287 | DOI | MR | Zbl

[25] B. A. Sevast'yanov, A. M. Zubkov, “Controlled branching processes”, Theory Probab. Appl., 19:1 (1974), 14–24 | DOI | MR | Zbl

[26] J. M. Steele, “The Bruss–Robertson inequality: elaborations, extensions, and applications”, Math. Appl. (Warsaw), 44:1 (2016), 3–16 | DOI | MR | Zbl

[27] D. Tasche, “On the second Borel–Cantelli lemma for strongly mixing sequences of events”, J. Appl. Probab., 34:2 (1997), 381–394 | DOI | MR | Zbl

[28] H. Terelius, K. H. Johansson, “Peer-to-peer gradient topologies in networks with churn”, IEEE Trans. Control Netw. Syst., 5:4 (2018), 2085–2095 | DOI | MR | Zbl

[29] V. A. Vatutin, “Asymptotic behavior of the non-extinction probability for a critical, branching process”, Theory Probab. Appl., 22:1 (1977), 140–146 | DOI | MR | Zbl

[30] V. A. Vatutin, A. M. Zubkov, “Branching processes. I”, J. Soviet Math., 39:1 (1987), 2431–2475 | DOI | MR | Zbl

[31] V. A. Vatutin, A. M. Zubkov, “Branching processes. II”, J. Soviet Math., 67:6 (1993), 3407–3485 | DOI | MR | Zbl

[32] J. M. Wirtz, Coalescent theory and Yule trees in time and space, PhD thesis, Univ. Köln, Köln, 2019, xii+91 pp.

[33] N. M. Yanev, “Conditions for degeneracy of $\varphi$-branching processes with random $\varphi$”, Theory Probab. Appl., 20:2 (1976), 421–428 | DOI | MR | Zbl

[34] A. M. Zubkov, “A degeneracy condition for a bounded branching process”, Math. Notes, 8:1 (1970), 472–477 | DOI | MR | Zbl

[35] A. M. Zubkov, “A degeneracy condition for bounded continuous-time branching processes”, Theory Probab. Appl., 17:2 (1973), 284–297 | DOI | MR | Zbl

[36] A. M. Zubkov, “Analogies between Galton–Watson processes and $\varphi$-branching processes”, Theory Probab. Appl., 19:2 (1975), 309–331 | DOI | MR | Zbl

[37] A. M. Zubkov, “Limiting distributions of the distance to the closest common ancestor”, Theory Probab. Appl., 20:3 (1976), 602–612 | DOI | MR | Zbl