## The Annals of Probability

### The survival probability of a critical multi-type branching process in i.i.d. random environment

#### Abstract

We study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton–Watson process in i.i.d. random environments by using limit theorems for products of positive random matrices. Under suitable assumptions, the survival probability is proportional to $1/\sqrt{n}$.

#### Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2946-2972.

Dates
Revised: November 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097643

Digital Object Identifier
doi:10.1214/17-AOP1243

Mathematical Reviews number (MathSciNet)
MR3846842

Zentralblatt MATH identifier
06964352

#### Citation

Le Page, E.; Peigné, M.; Pham, C. The survival probability of a critical multi-type branching process in i.i.d. random environment. Ann. Probab. 46 (2018), no. 5, 2946--2972. doi:10.1214/17-AOP1243. https://projecteuclid.org/euclid.aop/1535097643

#### References

• [1] Afanasyev, V. I. (1993). A limit theorem for a critical branching process in a random environment. Discrete Math. Appl. 5 45–58.
• [2] Athreya, K. B. and Karlin, S. (1971). On branching processes with random environments. I. Extinction probabilities. Ann. Math. Stat. 42 1499–1520.
• [3] Athreya, K. B. and Karlin, S. (1971). Branching processes with random environments. II. Limit theorems. Ann. Math. Stat. 42 1843–1858.
• [4] Bansaye, V. and Berestycki, J. (2009). Large deviations for branching processes in random environment. Markov Process. Related Fields 15 493–524.
• [5] Bougerol, Ph. and Lacroix, J. (1985). Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Basel.
• [6] Dyakonova, E. E. (1999). The asymptotics of the probability of nonextinction of a multidimensional branching process in a random environment. Discrete Math. Appl. 9 119–136.
• [7] Dyakonova, E. E., Geiger, J. and Vatutin, V. A. (2004). On the survival probability and a functional limit theorem for branching processes in random environment. Markov Process. Related Fields 10 289–306.
• [8] Dyakonova, E. E. and Vatutin, V. A. (2017). Multitype branching processes in random environment: Survival probability for the critical case. Teor. Veroyatn. Primen. 62 634–653.
• [9] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Stat. 31 457–469.
• [10] Geiger, J. and Kersting, G. (2000). The survival probability of a critical branching process in random environment. Teor. Veroyatn. Primen. 45 607–615.
• [11] Geiger, J., Kersting, G. and Vatutin, V. A. (2003). Limit theorems for subcritical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 39 593–620.
• [12] Grama, I., Le Page, É. and Peigné, M. (2014). On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains. Colloq. Math. 134 1–55.
• [13] Grama, I., Le Page, E. and Peigné, M. (2017). Conditional limit theorems for products of random matrices. Probab. Theory Related Fields 168 601–639.
• [14] Guivarc’h, Y. and Hardy, J. (1988). Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré Probab. Stat. 24 73–98.
• [15] Guivarc’h, Y., Le Page, E. and Liu, Q. (2003). Normalisation d’un processus de branchement critique dans un environnement aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 337 603–608.
• [16] Guivarc’h, Y. and Liu, Q. (2001). Asymptotic properties of branching processes in a random environment. C. R. Acad. Sci. Paris Sér. I Math. 332 339–344.
• [17] Hennion, H. (1997). Limit theorems for products of positive random matrices. Ann. Probab. 25 1545–1587.
• [18] Kaplan, N. (1974). Some results about multidimensional branching processes with random environments. Ann. Probab. 2 441–455.
• [19] Kozlov, M. V. (1976). On the asymptotic bahaviour of the probability of non-extinction for critical branching processes in a random environment. Theory Probab. Appl. XXI 791–804.
• [20] Le Page, É. (1982). Théorèmes limites pour les produits de matrices aléatoires. In Probability Measures on Groups (Oberwolfach, 1981). Lecture Notes in Math. 928 258–303. Springer, Berlin.
• [21] Pham, T. D. C. (2018). Conditioned limit theorems for products of positive random matrices. ALEA, Lat. Am. J. Probab. Math. Stat. 15 67–100.
• [22] Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Stat. 40 814–827.
• [23] Zubkov, A. M. (1994). Inequalities for the distribution of the numbers of simultaneous events. Survey Appl. Industry Math. Ser. Probab. Stat. 1 638–666.