Conditional limit theorems for intermediately subcritical branching processes in random environment

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1. [1] V. I. Afanasyev. Limit theorems for a conditional random walk and some applications. Diss. Cand. Sci., MSU, Moscow, 1980. 
2. [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. Zbl1045.60087MR1846044
3. [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. Zbl1262.60083MR2956209
4. [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. Zbl1075.60107MR2123206
5. [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. Zbl1080.60079MR2165338
6. [6] A. Agresti. On the extinction times of varying and random environment branching processes. J. Appl. Probab.12 (1975) 39–46. Zbl0306.60052MR365733
7. [7] K. B. Athreya and S. Karlin. On branching processes with random environments: I, II. Ann. Math. Stat. 42 (1971) 1499–1520, 1843–1858. Zbl0228.60033
8. [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. Zbl1245.60081MR2851050
9. [9] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. Zbl0938.60005MR1406564
10. [10] J. Bertoin and R. A. Doney. On conditioning a random walk to stay non-negative. Ann. Probab.22 (1994) 2152–2167. Zbl0834.60079MR1331218
11. [11] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. Zbl0667.26003MR898871
12. [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. Zbl1084.60062MR2118578
13. [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. Zbl1198.60045MR2673988
14. [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. Zbl1288.60107
15. [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. Zbl1175.60029MR2451576
16. [16] L. Chaumont. Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math.121 (1997) 377–403. Zbl0882.60074MR1465814
17. [17] B. Chauvin, A. Rouault and A. Wakolbinger. Growing conditioned trees. Stochastic Process. Appl.39 (1991) 117–130. Zbl0747.60077MR1135089
18. [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. Zbl0634.60072MR934535
19. [19] R. A. Doney. Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. verw. Gebiete 70 (1985) 351–360. Zbl0573.60063MR803677
20. [20] J. Geiger. Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Probab.36 (1999) 301–309. Zbl0942.60071MR1724856
21. [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. Zbl1038.60083MR1983172
22. [22] Y. Guivarc’h and Q. Liu. Propriétés asymptotiques des processus de branchement en environnement aléatoire. C. R. Math. Acad. Sci. Paris Sér. I332 (2001) 339–344. Zbl0988.60080
23. [23] O. Kallenberg. Stability of critical cluster fields. Math. Nachr.77 (1977) 7–43. Zbl0361.60058MR443078
24. [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. Zbl1126.60089MR2283329
25. [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. Zbl1213.60162MR2766343
26. [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. Zbl0840.60077MR1349164
27. [27] J. Neveu. Erasing a branching tree. Adv. Appl. Probab.18 (1986) 101–108. Zbl0613.60078MR868511
28. [28] W. L. Smith and W. E. Wilkinson. On branching processes in random environments. Ann. Math. Stat.40 (1969) 814–827. Zbl0184.21103MR246380
29. [29] H. Tanaka. Time reversal of random walks in one dimension. Tokyo J. Math.12 (1989) 159–174. Zbl0692.60052MR1001739
30. [30] V. A. Vatutin. A limit theorem for an intermediate subcritical branching process in a random environment. Theory Probab. Appl.48 (2004) 481–492. Zbl1068.60096MR2141345
31. [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. Zbl1079.60080MR2015453
32. [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. Zbl1091.60037MR2144298
33. [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. Zbl1114.60085MR2324164

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