×

The survival probability of a critical multi-type branching process in i.i.d. random environment. (English) Zbl 1445.60063

The authors study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments. Under suitable assumptions, including that the random environments admit so-called linear-fractional multidimensional generating functions, the authors obtain an optimal result, it is shown that the survival probability up to time \(n\) multiplied by \(\sqrt{n}\) converges to some positive limit as \(n\to\infty\). It is explained that the main difficulty which appears in the proofs is that the fluctuations of products of some random matrices have to be investigated, and there were no available results in this direction at that time. In case of general generating functions of the random environments, the authors also prove a weaker result.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
PDF BibTeX XML Cite
Full Text: DOI Euclid

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. · Zbl 0228.60032
[3] Athreya, K. B. and Karlin, S. (1971). Branching processes with random environments. II. Limit theorems. Ann. Math. Stat.42 1843–1858. · Zbl 0228.60033
[4] Bansaye, V. and Berestycki, J. (2009). Large deviations for branching processes in random environment. Markov Process. Related Fields15 493–524. · Zbl 1193.60098
[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. · Zbl 0969.60086
[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 Fields10 289–306. · Zbl 1078.60065
[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. · Zbl 1400.60111
[9] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Stat.31 457–469. · Zbl 0137.35501
[10] Geiger, J. and Kersting, G. (2000). The survival probability of a critical branching process in random environment. Teor. Veroyatn. Primen.45 607–615. · Zbl 0994.60095
[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. · Zbl 1038.60083
[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. · Zbl 1302.60057
[13] Grama, I., Le Page, E. and Peigné, M. (2017). Conditional limit theorems for products of random matrices. Probab. Theory Related Fields168 601–639. · Zbl 1406.60015
[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. · Zbl 1031.60087
[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. · Zbl 0903.60027
[18] Kaplan, N. (1974). Some results about multidimensional branching processes with random environments. Ann. Probab.2 441–455. · Zbl 0293.60078
[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. · Zbl 0384.60058
[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. · Zbl 1388.60032
[22] Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Stat.40 814–827. · Zbl 0184.21103
[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. · Zbl 0836.60014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.