Decomposable branching processes with two types of particles

V. A. Vatutin ; E. E. D'yakonova

Diskretnaya Matematika, Tome 30 (2018) no. 1, pp. 3-18.

Résumé

A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable $\Xi _{2},$ the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as $N\rightarrow \infty $) the conditional distribution of the amount of the first type particles in different generations given that either $\Xi _{2}=N$ or $\Xi _{2}>N.$

Détail

Keywords: decomposable branching process, total population size, limit theorem.

Bibliographie

[1] Afanasev V. I., “Funktsionalnye predelnye teoremy dlya razlozhimogo vetvyaschegosya protsessa s dvumya tipami chastits”, Diskretnaya matematika, 27:2 (2015), 22–44 ; Afanasyev V. I., “Functional limit theorems for the decomposable branching process with two types of particles”, Discrete Math. Appl., 26:2 (2016), 71–88 | DOI | MR

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

[3] Afanasev V. I., “Funktsionalnaya predelnaya teorema dlya razlozhimogo vetvyaschegosya protsessa s dvumya tipami chastits”, Matem. zametki, 103:3 (2018), 323–335

[4] Vatutin V. A., “Struktura razlozhimykh redutsirovannykh vetvyaschikhsya protsessov. I. Konechnomernye raspredeleniya”, Teoriya veroyatn. i ee primenen., 59:4 (2014), 667-692 ; Vatutin V. A., “The structure of decomposable reduced branching processes. I. Finite-dimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662 | DOI | MR | DOI | MR

[5] Vatutin V. A., “Struktura razlozhimykh redutsirovannykh vetvyaschikhsya protsessov. II. Funktsionalnye predelnye teoremy”, Teoriya veroyatn. i ee primenen., 60:1 (2015), 25–44 ; Vatutin V. A., “The structure of decomposable reduced branching processes. II. Functional limit theorems”, Theory Probab. Appl., 60:1 (2016), 103–119 | DOI | MR | DOI | MR

[6] Vatutin V. A., “Uslovnaya funktsionalnaya predelnaya teorema dlya razlozhimykh vetvyaschikhsya protsessov s dvumya tipami chastits”, Matem. zametki, 101:5 (2017), 669–683 | DOI | MR

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

[8] Vatutin V. A., Dyakonova E. E., “O vyrozhdenii razlozhimykh vetvyaschikhsya protsessov”, Diskretnaya matematika, 28:4 (2015), 26–37 ; Vatutin V. A., Dyakonova E. E., “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192 | DOI | DOI | MR

[9] Vatutin V. A. , Sagitov S. M., “Razlozhimyi kriticheskii vetvyaschiisya protsess s dvumya tipami chastits”, Veroyatnostnye zadachi diskretnoi matematiki, Tr. MIAN SSSR, 177, 1986, 3–20 ; Vatutin V. A., Sagitov S. M., “A decomposable critical branching process with two types of particles”, Proc. Steklov Inst. Math., 177, 1988, 1–19 | MR

[10] Gnedenko B. V., Kolmogorov A. N., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Gos. iz-vo tekhniko-teoreticheskoi literatury, Moskva, Leningrad, 1949 ; Gnedenko B. V., Kolmogorov A. N., Limit distributions for sums of independent random variables, Addison-Wesley, Reading, London, 1954 | MR | MR

[11] Foster J., Ney P., “Decomposable critical multi-type branching processes”, Invited paper for Mahalanobis Memorial symposium (Calcutta), Sanhya: the Indian J. Stat., Series A, 38 (1976), 28–37 | MR

[12] Foster J., Ney P., “Limit laws for decomposable critical branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 46 (1978), 13–43 | DOI | MR

[13] Ogura Y., “Asymptotic behavior of multitype Galton-Watson processes”, J. Math. Kyoto Univ., 15 (1975), 251–302 | DOI | MR

[14] Savin A. A., Chistyakov V. P., “Nekotorye teoremy dlya vetvyaschikhsya protsessov s neskolkimi tipami chastits”, Teoriya veroyatn. i ee primenen., 7:1 (1962), 95–104

[15] Sagitov S. M., “Mnogomernye predelnye teoremy dlya vetvyaschegosya protsessa s odnim tipom chastits”, Matem. zametki, 42:1 (1987), 157–165 | MR

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

[17] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, Moskva, 1985 ; Seneta E., Regularly Varying Functions, Springer-Verlag, Berlin, Heidelberg, New York, 1976 | MR | MR

[18] Kersting G., On the height profile of a conditioned Galton-Watson tree, 24 pages, 2011, arXiv: 1101.3656

[19] Kolchin V. F., Sluchainye otobrazheniya, Nauka, Moskva, 1984 ; Kolchin V. F., Random Mappings, Optimization Software, Inc., Publication Division, 1986 | MR | MR

[20] Bertoin J., Levy Processes, Cambrige University Press, Cambridge, 1996 | MR

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

[22] Zolotarev V. M., Odnomernye ustoichivye raspredeleniya, Nauka, Moskva, 1983 ; Zolotarev V. M., One-dimensional Stable Distributions, Transl. of Math. Monographs, 65, Amer. Math. Soc., Providence, 1986 | MR | MR

[23] Zubkov A. M., “Predelnoe povedenie razlozhimykh kriticheskikh vetvyaschikhsya protsessov s dvumya tipami chastits”, Teoriya veroyatn. i ee primenen., 27:2 (1982), 228–238 ; Zubkov A. M., “The limit behaviour of decomposable critical branching processes with two types of particles”, Theory Probab. Appl., 27:2 (1983), 235–237 | MR | DOI | MR

[24] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, Moskva, 1984 ; Feller W., An Introduction to Probability Theory and its Applications, v. II, John Wiley Sons, Inc., New York, London, Sydney, Toronto, 1971 | MR | MR