Small positive values for supercritical branching processes in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, p. 770-805

Branching Processes in Random Environment (BPREs) (Z n :n0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of (1Z n k|Z 0 =i), k,i as n. More precisely, we characterize the exponential decrease of (Z n =k|Z 0 =i) using a spine representation due to Geiger. We then provide some bounds for this rate of decrease. If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.

Les processus de branchement en environnement aléatoire (Z n :n0) sont une généralisation des processus de Galton Watson où à chaque génération, la reproduction est choisie de manière i.i.d. Dans le régime surcritique, ces processus survivent avec probabilité positive et croissent alors géométriquement. Ce papier considère l’événement rare où le processus prend des valeurs non nulles mais bornées en temps long. Nous décrivons ainsi le comportement asymptotique de P(1Z n k|Z 0 =i) quand n. Plus précisément, nous caractérisons la vitesse exponentielle àlaquelle (Z n =k|Z 0 =i) tend vers zéro en utilisant une représentation en épine due à Geiger. Nous donnons alors des bornes pour cette vitesse. Si la loi de reproduction est linéaire fractionnaire, la vitesse devient plus explicite et deux régimes apparaissent. Nous montrons par ailleurs que ces régimes affectent le comportement asymptotique de l’ancêtre commun le plus récent de la population en vie à l’instant n quand cette dernière est conditionnée à prendre de petites valeurs en temps long.

DOI : https://doi.org/10.1214/13-AIHP538
Classification:  60J80,  60K37,  60J05,  60F17,  92D25
Keywords: supercritical branching processes, random environment, large deviations, phase transitions
@article{AIHPB_2014__50_3_770_0,
     author = {Bansaye, Vincent and B\"oinghoff, Christian},
     title = {Small positive values for supercritical branching processes in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {770-805},
     doi = {10.1214/13-AIHP538},
     zbl = {06340408},
     mrnumber = {3224289},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_3_770_0}
}
Bansaye, Vincent; Böinghoff, Christian. Small positive values for supercritical branching processes in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 770-805. doi : 10.1214/13-AIHP538. http://www.numdam.org/item/AIHPB_2014__50_3_770_0/

[1] V. I. Afanasyev, C. Böinghoff, G. Kersting and V. A. Vatutin. Conditional limit theorems for intermediately subcritical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 602-627. | MR 3189086 | Zbl 1290.60083

[2] V. I. Afanasyev, C. Böinghoff, G. Kersting and V. A. Vatutin. Limit theorems for a weakly subcritical branching process in random environment. J. Theoret. Probab. 25 (2012) 703-732. | MR 2956209 | Zbl 1262.60083

[3] 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 2165338 | Zbl 1080.60079

[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 2123206 | Zbl 1075.60107

[5] A. Agresti. On the extinction times of varying and random environment branching processes. J. Appl. Probab. 12 (1975) 39-46. | MR 365733 | Zbl 0306.60052

[6] K. B. Athreya. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab. 4 (1994) 779-790. | MR 1284985 | Zbl 0806.60068

[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 0228.60033

[8] K. B. Athreya and P. E. Ney. Branching Processes. Dover, Mineola, NY, 2004. | MR 2047480 | Zbl 1070.60001

[9] V. Bansaye and J. Berestycki. Large deviations for branching processes in random environment. Markov Process. Related Fields 15 (2009) 493-524. | MR 2598126 | Zbl 1193.60098

[10] V. Bansaye and C. Böinghoff. Upper large deviations for branching processes in random environment with heavy tails. Electron. J. Probab. 16 (2011) 1900-1933. | MR 2851050 | Zbl 1245.60081

[11] C. Böinghoff. Branching processes in random environment. Ph.D. thesis, Goethe-Univ. Frankfurt/Main, 2010.

[12] C. 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 2673988 | Zbl 1198.60045

[13] F. M. Dekking. On the survival probability of a branching process in a finite state iid environment. Stochastic Process. Appl. 27 (1998) 151-157. | MR 934535 | Zbl 0634.60072

[14] K. Fleischmann and V. Vatutin. Reduced subcritical Galton-Watson processes in a random environment. Adv. in Appl. Probab. 31 (1999) 88-111. | MR 1699663 | Zbl 0938.60090

[15] K. Fleischmann and V. Wachtel. On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 201-225. | Numdam | MR 2500235 | Zbl 1175.60075

[16] J. Geiger. Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab. 36 (1999) 301-309. | MR 1724856 | Zbl 0942.60071

[17] 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 1983172 | Zbl 1038.60083

[18] Y. Guivarc'H and Q. Liu. Asymptotic properties of branching processes in random environment. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 339-344. | MR 1821473 | Zbl 0988.60080

[19] B. Hambly. On the limiting distribution of a supercritical branching process in random environment. J. Appl. Probab. 29 (1992) 499-518. | MR 1174427 | Zbl 0758.60089

[20] C. Huang and Q. Liu. Moments, moderate and large deviations for a branching process in a random environment. Stochastic Process. Appl. 122 (2010) 522-545. | MR 2868929 | Zbl 1242.60087

[21] C. Huang and Q. Liu. Convergence in L p and its exponential rate for a branching process in a random environment, 2011. Avialable at http://arxiv.org/abs/1011.0533. | MR 3275856

[22] 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 1349164 | Zbl 0840.60077

[23] M. Hutzenthaler. Supercritical branching diffusions in random environment. Electron. Commun. Probab. 16 (2011) 781-791. | MR 2868599 | Zbl 1243.60080

[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 2283329 | Zbl 1126.60089

[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 2766343 | Zbl 1213.60162

[26] E. Marchi. When is the product of two concave functions concave? Int. J. Math. Game Theory Algebra 19 (2010) 165-172. | MR 2730404 | Zbl 1211.26013

[27] J. Neveu. Erasing a branching tree. Adv. Apl. Probab. suppl. (1986) 101-108. | MR 868511 | Zbl 0613.60078

[28] A. Rouault. Large deviations and branching processes. In Proceedings of the 9th International Summer School on Probability Theory and Mathematical Statistics (Sozopol, 1997) 15-38. Pliska Stud. Math. Bulgar. 13. Bulgarian Academy of Sciences, Sofia, 2000. | MR 1800359 | Zbl 0967.60083

[29] W. L. Smith and W. E. Wilkinson. On branching processes in random environments. Ann. Math. Stat. 40 (1969) 814-824. | MR 246380 | Zbl 0184.21103

[30] V. A. Vatutin and V. Wachtel. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009) 177-217. | MR 2449127 | Zbl 1158.60014