Asymptotic properties of supercritical branching processes in random environments

Abstract

We consider a supercritical branching process (Z_n) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale W_n=Z_n/\mathbb{E}[Zn|\xi ], the convergence rates of W-W_n(by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in L^P, and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Z_n).

Keywords

Branching process / random environment / large deviation / moderate deviation / central limit theorem / moment / weighted moment / convergence rate

References

[1] Afanasyev V I, Geiger J, Kersting G, Vatutin V A. Criticality for branching processes in random environment. Ann Probab, 2005, 33(2): 645-673

[2] Afanasyev V I, Geiger J, Kersting G, Vatutin V A. Functional limit theorems for strongly subcritical branching processes in random environment. Stochastic Process Appl, 2005, 115(10): 1658-1676

[3] Alsmeyer G, Rösler U. On the existence of ϕ-moments of the limit of a normalized supercritical Galton-Watson process. J Theoret Probab, 2004, 17(4): 905-928

[4] Asmussen S. Convergence rates for branching processes. Ann Probab, 1976, 4: 139-146

[5] Athreya K B. Large deviation rates for branching processes—I. Single type case. Ann Appl Probab, 1994, 4: 779-790

[6] Athreya K B, Karlin S. On branching processes in random environments I. Ann Math Statist, 1971, 42: 1499-1520

[7] Athreya K B, Karlin S. On branching processes in random environments II. Ann Math Statist, 1971, 42: 1843-1858

[8] Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972

[9] Bansaye V, Berestycki J. Large deviations for branching processes in random environment. Markov Process Related Fields, 2009, 15: 493-524

[10] Bansaye V, Böinghoff C. Upper large deviations for branching processes in random environment with heavy tails. Preprint, 2010

[11] Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes I: The Galton-Watson processes. Adv Appl Probab, 1974, 6: 711-731

[12] Böinghoff C, Dyakonova E E, Kersting G, Vatutin V A. Branching processes in random environment which extinct at a given moment. Markov Process Related Fields, 2010, 16(2): 329-350

[13] Böinghoff C, Kersting G. Upper large deviations of branching processes in a random environment-Offspring distributions with geometrically bounded tails. Stochastic Process Appl, 2010, 120: 2064-2077

[14] Dembo A, Zeitouni O. Large deviations Techniques and Applications. New York: Springer, 1998

[15] Fearn D H. Galton-Watson processes with generation dependence. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ California, Berkeley, Calif, 1970/1971), Vol IV: Biology and Health, Berkeley, Calif. Berkeley: Univ California Press, 1972, 159-172

[16] Guivarc’h Y, Liu Q. Propriétés asymptotiques des processus de branchement en environnement aléatoire. C R Acad Sci Paris, Ser I, 2001, 332: 339-344

[17] Hambly B. On the limit distribution of a supercritical branching process in a random environment. J Appl Probab, 1992, 29: 499-518

[18] Harris T E. The Theory of Branching Processes. Berlin: Springer, 1963

[19] Heyde C C. A rate of convergence result for the super-critical Galton-Watson process. J Appl Probab, 1970, 7: 451-454

[20] Heyde C C. Some central limit analogues for super-critical Galton-Watson process. J Appl Probab, 1971, 8: 52-59

[21] Heyde C C, Brown B M. An invariance principle and some convergence rate results for branching processes. Z Wahrscheinlichkeitstheorie verw Geb, 1971, 20: 189-192

[22] Huang C, Liu Q. Moments, moderate and large deviations for a branching process in a random environment. Stochastic Process Appl, 2012, 122: 522-545

[23] Huang C, Liu Q. Convergence rates for a supercritical branching process in a random environment. Markov Process Related Fields (to appear)

[24] Huang C, Liu Q. Convergence in L^P and its exponential rate for a branching process in a random environment. Preprint, 2014

[25] Iksanov A. On some moments of the limit random variable for a normalized supercritical Galton-Watson process. In: Velle L R, ed. Focus on Probability Theory. New York: Nova Science Publishers, Inc, 2006, 151-158

[26] Jagers P. Galton-Watson processes in varying environments. J Appl Probab, 1974, 11: 174-178

[27] Kozlov M V. On large deviations of branching processes in a random environment: geometric distribution of descendants. Discrete Math Appl, 2006, 16: 155-174

[28] Li Y, Hu Y, Liu Q. Weighted moments for a supercritical branching process in a varying or random environment. Sci China Ser A: Math, 2011, 54(7): 1437-1444

[29] Liang X, Liu Q. Weighted moments for the limit of a normalized supercritical Galton-Watson process. C R Math Acad Sci Paris, 2013, 351(19-20): 769-773

[30] Liang X, Liu Q. Weighted moments of the limit of a branching process in a random environment. Proc Steklov Inst Math, 2013, 282(1): 127-145

[31] Liu Q. The exact Hausdorff dimension of a branching set. Probab Theory Related Fields, 1996, 104: 515-538

[32] Liu Q. Local dimensions of the branching measure on a Galton-Watson tree. Ann Inst Henri Poincaré, Probabilitiés et Statistique, 2001, 37: 195-222

[33] Ney P E, Vidyashanker A N. Harmonic moments and large deviation rates for supercritical branching process. Ann Appl Probab, 2003, 13: 475-489

[34] Smith W L, Wilkinson W E. On branching processes in random environments. Ann Math Statist, 1969, 40: 814-827

[35] Tanny D. Limit theorems for branching processes in a random environment. Ann Probab, 1977, 5: 100-116

[36] Tanny D. A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stochastic Process Appl, 1988, 28: 123-139

[37] Vatutin V A, Liu Q. Critical branching process with two types of particles evolving in asynchronous random environments. Theory Probab Appl, 2013, 57(2): 279-305

[38] Wang H, Gao Z, Liu Q. Central limit theorems for a branching process i<?Pub Caret?>n a random environment. Statist Probab Lett, 2011, 81: 539-547

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg