Conditional limit theorems for intermediately subcritical branching processes in random environment

Afanasyev, V. I.; Böinghoff, Ch.; Kersting, G.; Vatutin, V. A.

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 602-627.

Nous considérons un processus de branchement dans un environnement aléatoire dont la distribution des enfants des individus varie aléatoirement de façon indépendante d'une génération à l'autre. Dans le régime sous critique, une transition de phase apparaît. Cet article est consacré à l'étude de la région proche de la transition. Nous étudions le comportement asymptotique de la probabilité de survie ainsi que la taille de la population et la forme de l'environnement aléatoire sous la condition de non-extinction. Nous montrons finalement que conditionnée à la non-extinction, la population alterne des périodes de petite et de grande taille. Ce type de comportement apparaît sous la mesure moyennée uniquement dans ce régime sous critique proche de la transition.

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment conditioned on non-extinction is examined. Finally we show that conditioned on non-extinction periods of small and large population sizes alternate. This kind of ‘bottleneck' behavior appears under the annealed approach only in the intermediately subcritical case.

Classification : 60J80, 60K37, 60G50, 60F17
Mots clés : branching process, random environment, random walk, change of measure, survival probability, functional limit theorem, tree

[1] V. I. Afanasyev. Limit theorems for a conditional random walk and some applications. Diss. Cand. Sci., MSU, Moscow, 1980.

[2] V. I. Afanasyev. Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment. Discrete Math. Appl. 11 (2001) 105-131. | MR | Zbl

[3] V. I. Afanasyev, Ch. Böinghoff, G. Kersting and V. A. Vatutin. Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Probab. 25 (2012) 703-732. | MR | Zbl

[4] V. I. Afanasyev, J. Geiger, G. Kersting and V. A. Vatutin. Criticality for branching processes in random environment. Ann. Probab. 33 (2005) 645-673. | MR | Zbl

[5] V. I. Afanasyev, J. Geiger, G. Kersting and V. A. Vatutin. Functional limit theorems for strongly subcritical branching processes in random environment. Stochastic Process. Appl. 115 (2005) 1658-1676. | MR | Zbl

[6] A. Agresti. On the extinction times of varying and random environment branching processes. J. Appl. Probab. 12 (1975) 39-46. | MR | Zbl

[7] K. B. Athreya and S. Karlin. On branching processes with random environments: I, II. Ann. Math. Stat. 42 (1971) 1499-1520, 1843-1858. | Zbl

[8] V. Bansaye and Ch. Böinghoff. Upper large deviations for branching processes in random environment with heavy tails. Electron. J. Probab. 16 (2011) 1900-1933. | MR | Zbl

[9] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[10] J. Bertoin and R. A. Doney. On conditioning a random walk to stay non-negative. Ann. Probab. 22 (1994) 2152-2167. | MR | Zbl

[11] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl

[12] M. Birkner, J. Geiger and G. Kersting. Branching processes in random environment - A view on critical and subcritical cases. In Proceedings of the DFG-Schwerpunktprogramm Interacting Stochastic Systems of High Complexity 269-291. Springer, Berlin, 2005. | MR | Zbl

[13] Ch. Böinghoff and G. Kersting. Upper large deviations of branching processes in a random environment - Offspring distributions with geometrically bounded tails. Stochastic Process. Appl. 120 (2010) 2064-2077. | MR | Zbl

[14] Ch. Böinghoff and G. Kersting. Simulations and a conditional limit theorem for intermediately subcritcal branching processes in random environment. Preprint, 2012. Available at arXiv:1209.1274. | Zbl

[15] F. Caravenna and L. Chaumont. Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 170-190. | Numdam | MR | Zbl

[16] L. Chaumont. Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377-403. | MR | Zbl

[17] B. Chauvin, A. Rouault and A. Wakolbinger. Growing conditioned trees. Stochastic Process. Appl. 39 (1991) 117-130. | MR | Zbl

[18] F. M. Dekking. On the survival probability of a branching process in a finite state i.i.d. environment. Stochastic Process. Appl. 27 (1988) 151-157. | MR | Zbl

[19] R. A. Doney. Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. verw. Gebiete 70 (1985) 351-360. | MR | Zbl

[20] J. Geiger. Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab. 36 (1999) 301-309. | MR | Zbl

[21] J. Geiger, G. Kersting and V. A. Vatutin. Limit theorems for subcritical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 39 (2003) 593-620. | Numdam | MR | Zbl

[22] Y. Guivarc'H and Q. Liu. Propriétés asymptotiques des processus de branchement en environnement aléatoire. C. R. Math. Acad. Sci. Paris Sér. I 332 (2001) 339-344. | Zbl

[23] O. Kallenberg. Stability of critical cluster fields. Math. Nachr. 77 (1977) 7-43. | MR | Zbl

[24] M. V. Kozlov. On large deviations of branching processes in a random environment: Geometric distribution of descendants. Discrete Math. Appl. 16 (2006) 155-174. | MR | Zbl

[25] M. V. Kozlov. On large deviations of strictly subcritical branching processes in a random environment with geometric distribution of progeny. Theory Probab. Appl. 54 (2010) 424-446. | MR | Zbl

[26] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of $LlogL$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125-1138. | MR | Zbl

[27] J. Neveu. Erasing a branching tree. Adv. Appl. Probab. 18 (1986) 101-108. | MR | Zbl

[28] W. L. Smith and W. E. Wilkinson. On branching processes in random environments. Ann. Math. Stat. 40 (1969) 814-827. | MR | Zbl

[29] H. Tanaka. Time reversal of random walks in one dimension. Tokyo J. Math. 12 (1989) 159-174. | MR | Zbl

[30] V. A. Vatutin. A limit theorem for an intermediate subcritical branching process in a random environment. Theory Probab. Appl. 48 (2004) 481-492. | MR | Zbl

[31] V. A. Vatutin and E. E. Dyakonova. Galton-Watson branching processes in random environment. I: Limit theorems. Theory Probab. Appl. 48 (2004) 314-336. | MR | Zbl

[32] V. A. Vatutin and E. E. Dyakonova. Galton-Watson branching processes in random environment. II: Finite-dimensional distributions. Theory Probab. Appl. 49 (2005) 275-308. | MR | Zbl

[33] V. A. Vatutin and E. E. Dyakonova. Branching processes in random environment and the bottlenecks in the evolution of populations. Theory Probab. Appl. 51 (2007) 189-210. | MR | Zbl