Geodesic
A Conditional Functional Limit Theorem for Decomposable Branching Processes with Two Types of Particles
Matematičeskie zametki, Tome 101 (2017) no. 5, pp. 669-683.

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Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time $Nt$, $0\leq t\infty$, given that the number of particles of the first type appearing in the process during its evolution is $N$.
Keywords: decomposable branching process, total size of the population, functional limit theorem. 
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V. A. Vatutin. A Conditional Functional Limit Theorem for Decomposable Branching Processes with Two Types of Particles. Matematičeskie zametki, Tome 101 (2017) no. 5, pp. 669-683. http://geodesic.mathdoc.fr/item/MZM_2017_101_5_a2/

[1] R. S. Slack, “A branching process with mean one and possibly infinite variance”, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 9:2 (1968), 139–145 | DOI | MR | Zbl

[2] E. Seneta, Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | MR | Zbl

[3] A. M. Zubkov, “Predelnoe povedenie razlozhimykh kriticheskikh vetvyaschikhsya protsessov s dvumya tipami chastits”, Teoriya veroyatn. i ee primen., 27:2 (1982), 228–238 | MR | Zbl

[4] V. A. Vatutin, S. M. Sagitov, “Razlozhimyi kriticheskii vetvyaschiisya protsess s dvumya tipami chastits”, Veroyatnostnye zadachi diskretnoi matematiki, Tr. MIAN SSSR, 177, 1986, 3–20 | MR | Zbl

[5] C. Smadi, V. A. Vatutin, “Reduced two-type decomposable critical branching processes with possibly infinite variance”, Markov Process. Related Fields, 22:2 (2016), 311–358 | MR

[6] B. A. Sevastyanov, Vetvyaschiesya protsessy, Nauka, M., 1971 | MR | Zbl

[7] K. B. Athreya, P. E. Ney, Branching Processes, Grundlehren Math. Wiss., 196, Springer-Verlag, Berlin, 1972 | MR | Zbl

[8] V. A. Vatutin, “Struktura razlozhimykh redutsirovannykh vetvyaschikhsya protsessov. I. Konechnomernye raspredeleniya”, Teoriya veroyatn. i ee primen., 59:4 (2014), 667–692 | DOI

[9] V. A. Vatutin, “Struktura razlozhimykh redutsirovannykh vetvyaschikhsya protsessov. II. Funktsionalnye predelnye teoremy”, Teoriya veroyatn. i ee primen., 60:1 (2015), 25–44 | DOI | MR

[10] V. A. Vatutin, E. E. Dyakonova, “Razlozhimye vetvyaschiesya protsessy s fiksirovannym momentom vyrozhdeniya”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki, Tr. MIAN, 290, MAIK, M., 2015, 114–135 | DOI

[11] V. A. Vatutin, E. E. Dyakonova, “O vyrozhdenii razlozhimykh vetvyaschikhsya protsessov”, Diskret. matem., 27:4 (2015), 26–37 | DOI | MR | Zbl

[12] M. Drmota, V. Vatutin, “Limiting distributions in branching processes with two types of particles”, Classical and Modern Branching Processes, IMA Vol. Math. Appl., 84, Springer, New York, 1987, 89–110 | DOI | MR | Zbl

[13] V. I. Afanasev, “Funktsionalnye predelnye teoremy dlya razlozhimogo vetvyaschegosya protsessa s dvumya tipami chastits”, Diskret. matem., 27:2 (2015), 22–44 | DOI | MR

[14] V. I. Afanasev, “O razlozhimom vetvyaschemsya protsesse s dvumya tipami chastits”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Tr. MIAN, 294, MAIK, M., 2016, 7–19 | DOI

[15] A. Grimvall, “On the convergence of sequences of branching processes”, Ann. Probab., 2:6 (1974), 1027–1045 | DOI | MR | Zbl

[16] J. Lamperti, “Limiting distributions for branching processes”, 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability Vol. II. Contributions to Probability Theory. Part 2, Univ. California Press, Berkeley, CA, 1967, 225–241 | MR | Zbl

[17] P. Billingsley, Convergence of Probability Measures, John Wiley Sons, New York, 1999 | MR | Zbl

[18] P. Billingsli, Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR | Zbl

[19] Zh. Zhakod, A. N. Shiryaev, Predelnye teoremy dlya sluchainykh protsessov, T. 1, Fizmatlit, M., 1994 | MR | Zbl

[20] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 2, Mir, M., 1984 | MR | Zbl

[21] T. Duquesne, “A limit theorem for the contour process of conditioned Galton–Watson trees”, Ann. Probab., 31:2 (2003), 996–1027 | MR | Zbl

[22] I. Kortchemski, “A simple proof of Duquesne's theorem on contour processes of conditioned Galton–Watson trees”, Séminaire de Probabilités XLV, Lecture Notes in Math., 2078, Springer-Verlag, Berlin, 2013, 537–558 | DOI | MR | Zbl

[23] T. Duquesne, J.-F. Le Gall, “Random Trees, Lévy Processes and Spatial Branching Processes”, Astérisque, 281 (2002), Soc. Math. France, Paris | MR | Zbl

[24] V. F. Kolchin, Sluchainye otobrazheniya, Teoriya veroyatnostei i matematicheskaya statistika, Nauka, M., 1984 | MR | Zbl