The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

V. A. Vatutin ; E. E. D'yakonova

Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 163-177.

Résumé

The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant $n$ of time for a process that started at the zero instant of time from one particle of any type is of the order of $\lambda^{n}n^{-1/2}$, where $\lambda \in (0,1)$ is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.

Détail

Keywords: branching process, random environment, survival probability, intermediately subcritical process, change of measures.

Bibliographie

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

[2] G. Kersting, V. Vatutin, Discrete Time Branching Processes in Random Environment, John Wiley Sons, London, 2017 | Zbl

[3] E. E. Dyakonova, “Dokriticheskii razlozhimyi vetvyaschiisya protsess v smeshannoi srede”, Diskret. matem., 30:2 (2018), 14–26 | DOI | Zbl

[4] E. E. Dyakonova, “Predelnaya teorema dlya mnogotipnogo kriticheskogo vetvyaschegosya protsessa, evolyutsioniruyuschego v sluchainoi srede”, Diskret. matem., 27:1 (2015), 44–58 | DOI | MR | Zbl

[5] E. E. Dyakonova, “Mnogotipnye dokriticheskie vetvyaschiesya protsessy v sluchainoi srede”, Vetvyaschiesya protsessy, sluchainye bluzhdaniya i smezhnye voprosy, Tr. MIAN, 282, MAIK «Nauka/Interperiodika», M., 2013, 87–97 | DOI | MR

[6] V. Vatutin, E. Dyakonova, P. Jagers, S. Sagitov, “A decomposable branching process in a Markovian environment”, Int. J. Stoch. Anal., 2012 (2012), Article ID 694285 | MR | Zbl

[7] E. E. Dyakonova, “Mnogotipnye vetvyaschiesya protsessy, evolyutsioniruyuschie v markovskoi srede”, Diskret. matem., 24:3 (2012), 130–151 | DOI | MR | Zbl

[8] E. E. Dyakonova, “Mnogotipnye vetvyaschiesya protsessy Galtona–Vatsona v markovskoi sluchainoi srede”, Teoriya veroyatn. i ee primen., 56:3 (2011), 592–601 | DOI | MR

[9] V. A. Vatutin, E. E. Dyakonova, “Asimptoticheskie svoistva mnogotipnykh kriticheskikh vetvyaschikhsya protsessov, evolyutsioniruyuschikh v sluchainoi srede”, Diskret. matem., 22:2 (2010), 22–40 | DOI | MR | Zbl

[10] E. Dyakonova, “On subcritical multi-type branching process in random environment”, Fifth Colloquium on Mathematics and Computer Science, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008, 397–404 | MR | Zbl

[11] E. E. Dyakonova, “Kriticheskie mnogotipnye vetvyaschiesya protsessy v sluchainoi srede”, Diskret. matem., 19:4 (2007), 23–41 | DOI | MR | Zbl

[12] V. Vatutin, Q. Liu, “Branching processes evolving in asynchronous environments”, Proceedings 59-th ISI World Statistics Congress, 25–30 August 2013, Hong Kong, International Statistical Institute, The Hague, The Netherlands, 2013, 1744–1749 http://2013.isiproceedings.org/Files/STS033-P3-S.pdf

[13] E. Le Page, M. Peigné, C. Pham, “The survival probability of a critical multi-type branching process in i.i.d. random environment”, Ann. Probab., 46:5 (2018), 2946–2972 | DOI | MR | Zbl

[14] V. A. Vatutin, E. E. Dyakonova, “Mnogotipnye vetvyaschiesya protsessy v sluchainoi srede: veroyatnost nevyrozhdeniya v kriticheskom sluchae”, Teoriya veroyatn. i ee primen., 62:4 (2017), 634–653 | DOI | Zbl

[15] V. Vatutin, V. Wachtel, “Multi-type subcritical branching processes in a random environment”, Adv. in Appl. Probab., 50:A (2018), 281–289 | DOI | MR

[16] T. D. C. Pham, “Conditional limit theorems for products of positive random matrices”, ALEA Lat. Am. J. Probab. Math. Stat., 15:1 (2018), 67–100 | DOI | MR | Zbl

[17] V. I. Afanasyev, Ch. Boeinghoff, G. Kersting, V. A. Vatutin, “Conditional limit theorems for intermediately subcritical branching processes in random environment”, Ann. Inst. Henri Poincaré Probab. Stat., 50:2 (2014), 602–627 | DOI | MR | Zbl

[18] D. Buraczewski, E. Damek, Y. Guivarc'h, S. Mentemeier, “On multidimensional Mandelbrot's cascades”, J. Difference Equ. Appl., 20:11 (2014), 1523–1567 | DOI | MR | Zbl

[19] J. F. Collamore, S. Mentemeier, “Large excursions and conditioned laws for recursive sequences generated by random matrices”, Ann. Probab., 46:4 (2018), 2064–2120 | DOI | MR | Zbl

[20] 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

[21] H. Furstenberg, H. Kesten, “Products of random matrices”, Ann. Math. Statist., 31:2 (1960), 457–469 | DOI | MR | Zbl