Multitype branching processes in random environment: survival probability for the critical case

V. A. Vatutin   ; E. E. D'yakonova

Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 4, pp. 634-653

Résumé

We investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in an environment generated by a sequence of independent identically distributed random variables. Under fairly general assumptions on the form of the offspring generating functions of particles, we show that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of type $i$ is equivalent to $\beta_in^{-1/2}$, where $\beta_i$ is a positive constant. This assertion essentially generalizes a number of previously known results.

Détail

Keywords: branching processes, random environment, survival probability, change of measure.

Bibliographie

[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] K. B. Athreya, S. Karlin, “On branching processes with random environments. I. Extinction probabilities”, Ann. Math. Statist., 42:5 (1971), 1499–1520 | DOI | MR | Zbl

[3] Ch. Böinghoff, E. E. Dyakonova, G. Kersting, V. A. Vatutin, “Branching processes in random environment which extinct at a given moment”, Markov Process. Related Fields, 16:2 (2010), 329–350 | MR | Zbl

[4] Ph. Bougerol, J. Lacroix, Products of random matrices with applications to Schrödinger operators, Progr. Probab. Statist., 8, Birkhäuser Boston, Inc., Boston, MA, 1985, xii+283 pp. | DOI | MR | Zbl

[5] E. E. Dyakonova, “Asymptotic behaviour of the probability of non-extinction for a multi-type branching process in a random environment”, Discrete Math. Appl., 9:2 (1999), 119–136 | DOI | DOI | MR | Zbl

[6] E. E. Dyakonova, “Critical multitype branching processes in a random environment”, Discrete Math. Appl., 17:6 (2007), 587–606 | DOI | DOI | MR | Zbl

[7] V. Vatutin, E. Dyakonova, “Part to survival for the critical branching processes in a random environment”, J. Appl. Probab., 54:2 (2017), 588–602 ; arXiv: 1603.03199 | DOI | MR

[8] E. E. Dyakonova, “Limit theorem for multitype critical branching process evolving in random environment”, Discrete Math. Appl., 25:3 (2015), 137–147 | DOI | DOI | MR | Zbl

[9] E. E. Dyakonova, “Redutsirovannye mnogotipnye kriticheskie vetvyaschiesya protsessy v sluchainoi srede”, Diskret. matem., 28:4 (2016), 58–79 | DOI

[10] E. E. Dyakonova, J. Geiger, V. A. Vatutin, “On the survival probability and a functional limit theorem for branching processes in random environment”, Markov Process. Related Fields, 10:2 (2004), 289–306 | MR | Zbl

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

[12] J. Geiger, G. Kersting, “The survival probability of a critical branching process in a random environment”, Teoriya veroyatn. i ee primen., 45:3 (2000), 607–615 ; Theory Probab. Appl., 45:3 (2001), 517–525 | DOI | MR | Zbl | DOI

[13] I. Grama, É. Le Page, M. Peigné, “Conditioned limit theorems for products of random matrices”, Probab. Theory Related Fields, 168:3-4 (2017), 601–639 ; arXiv: 1411.0423v2 | DOI | MR | Zbl

[14] M. V. Kozlov, “On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment”, Theory Probab. Appl., 21:4 (1977), 791–804 | DOI | MR | Zbl

[15] E. Le Page, M. Peigne, C. Pham, The survival probability of a critical multi-type branching process in i.i.d. random environment https://hal-univ-orleans.archives-ouvertes.fr/hal-01376091v1

[16] W. L. Smith, W. E. Wilkinson, “On branching processes in random environment”, Ann. Math. Statist., 40:3 (1969), 814–827 | DOI | MR | Zbl

[17] V. Vatutin, “A refinement of limit theorems for the critical branching processes in random environment”, Workshop on branching processes and their applications, Lect. Notes Stat. Proc., 197, Springer, Berlin, 2010, 3–19 | DOI | MR

[18] V. A. Vatutin, E. E. Dyakonova, “Asymptotic properties of multitype critical branching processes evolving in a random environment”, Discrete Math. Appl., 20:2 (2010), 157–177 | DOI | DOI | MR | Zbl

[19] V. A. Vatutin, “Total population size in critical branching processes in a random environment”, Math. Notes, 91:1 (2012), 12–21 | DOI | DOI | MR | Zbl

[20] V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242 | DOI | MR | Zbl

[21] V. Vatutin, E. Dyakonova, P. Jagers, S. Sagitov, “A decomposable branching process in a Markovian environment”, Int. J. Stoch. Anal., 2012 (2012), 694285, 24 pp. | DOI | MR | Zbl

[22] V. Vatutin, Quansheng Liu, “Branching processes evolving in asynchronous environments”, Proceedings of the 59-th ISI world statistics congress (Hong Kong, 2013), International Statistical Institute, The Hague, The Netherlands, 2013, 1744–1749 2013.isiproceedings.org/Files/STS033-P3-S.pdf

[23] V. Vatutin, Quansheng Liu, “Limit theorems for decomposable branching processes in a random environment”, J. Appl. Probab., 52:3 (2015), 877–893 | DOI | MR | Zbl

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

[25] V. A. Vatutin, V. Wachtel, “Sudden extinction of a critical branching process in a random environment”, Theory Probab. Appl., 54:3 (2010), 466–484 | DOI | DOI | MR | Zbl

[26] V. Vatutin, “Subcritical branching processes in random environment”, Branching processes and their applications (Badajoz, Spain, 2015), Lect. Notes Stat., 219, Springer, Cham, 2016, 97–115 | DOI | MR | Zbl

[27] C. Pham, Conditional limit theorems for products of positive random matrices, 2017, arXiv: 1703.04949v1