Evolution of branching processes in a~random environment

V. A. Vatutin ; E. E. Dyakonova ; S. Sagitov

Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 231-256.

Résumé

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

Bibliographie

[1] Afanasev V.I., Predelnye teoremy dlya uslovnogo sluchainogo bluzhdaniya i nekotorye primeneniya, Dis. ... kand. fiz.-mat. nauk, MGU, M., 1980

[2] Afanasev V.I., “Predelnaya teorema dlya kriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 5:1 (1993), 45–58 | MR | Zbl

[3] Afanasev V.I., “Novaya predelnaya teorema dlya kriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 9:3 (1997), 52–67 | DOI | MR | Zbl

[4] Afanasev V.I., “Predelnye teoremy dlya umerenno dokriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 10:1 (1998), 141–157 | DOI | MR | Zbl

[5] Afanasev V.I., “O maksimume kriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 11:2 (1999), 86–102 | DOI | MR | Zbl

[6] Afanasev V.I., “Predelnye teoremy dlya promezhutochno dokriticheskogo i strogo dokriticheskogo vetvyaschikhsya protsessov v sluchainoi srede”, Diskret. matematika, 13:1 (2001), 132–157 | DOI | Zbl

[7] Afanasev V.I., “Funktsionalnaya predelnaya teorema dlya kriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 13:4 (2001), 73–91 | DOI | MR | Zbl

[8] Afanasyev V.I., Böinghoff C., Kersting G., Vatutin V.A., “Limit theorems for weakly subcritical branching processes in random environment”, J. Theor. Probab., 25 (2012), 703–732 | DOI | MR | Zbl

[9] Afanasyev V.I., Böinghoff C., Kersting G., Vatutin V.A., “Conditional limit theorems for intermediately subcritical branching processes in random environment”, Ann. Inst. Henri Poincaré. Probab. Stat. (to appear) , arXiv: 1108.2127 [math.PR]

[10] Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Criticality for branching processes in random environment”, Ann. Probab., 33 (2005), 645–673 | DOI | MR | Zbl

[11] Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Functional limit theorems for strongly subcritical branching processes in random environment”, Stoch. Processes Appl., 115 (2005), 1658–1676 | DOI | MR | Zbl

[12] Albeverio S., Kozlov M.V., “O vozvratnosti i tranzientnosti zavisyaschikh ot sostoyaniya vetvyaschikhsya protsessov v sluchainoi srede”, Teoriya veroyatn. i ee primen., 48:4 (2003), 641–660 | DOI | MR | Zbl

[13] Athreya K.B., Karlin S., “On branching processes with random environments. I: Extinction probabilities”, Ann. Math. Stat., 42 (1971), 1499–1520 | DOI | MR | Zbl

[14] Athreya K.B., Karlin S., “Branching processes with random environments. II: Limit theorems”, Ann. Math. Stat., 42 (1971), 1843–1858 | DOI | MR | Zbl

[15] Athreya K.B., Ney P.E., Branching processes, Springer, Berlin, 1972 | MR | Zbl

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

[17] Birkner M., Geiger J., Kersting G., “Branching processes in random environment—A view on critical and subcritical cases”, Interacting stochastic systems, eds. J.-D. Deuschel, A. Greven, Springer, Berlin, 2005, 269–291 | DOI | MR

[18] Borovkov K.A., Vatutin V.A., “Reduced critical branching processes in random environment”, Stoch. Processes Appl., 71 (1997), 225–240 | DOI | MR | Zbl

[19] Böinghoff C., Dyakonova E.E., Kersting G., Vatutin V.A., “Branching processes in random environment which extinct at a given moment”, Markov Processes Relat. Fields, 16 (2010), 329–350 | MR | Zbl

[20] Dekking F.M., “On the survival probability of a branching process in a finite state i.i.d. environment”, Stoch. Processes Appl., 27 (1988), 151–157 | DOI | MR

[21] Doney R.A., “Conditional limit theorems for asymptotically stable random walks”, Z. Wahrscheinlichkeitstheor. verwandte Geb., 70 (1985), 351–360 | DOI | MR | Zbl

[22] Doney R.A., “Spitzer's condition and ladder variables in random walks”, Probab. Theory Relat. Fields, 101 (1995), 577–580 | DOI | MR | Zbl

[23] D'Souza J.C., Hambly B.M., “On the survival probability of a branching process in a random environment”, Adv. Appl. Probab., 29 (1997), 38–55 | DOI | MR | Zbl

[24] Durrett R., “Conditioned limit theorems for some null recurrent Markov processes”, Ann. Probab., 6 (1978), 798–828 | DOI | MR | Zbl

[25] Dyakonova E.E., “Ob asimptotike veroyatnosti nevyrozhdeniya mnogomernogo vetvyaschegosya protsessa v sluchainoi srede”, Diskret. matematika, 11:1 (1999), 113–128 | DOI | MR | Zbl

[26] Dyakonova E.E., “On multitype branching processes in a random environment”, J. Math. Sci., 111:3 (2002), 3537–3540 | DOI | MR | Zbl

[27] Dyakonova E.E., “Kriticheskie mnogotipnye vetvyascheesya protsessy v sluchainoi srede”, Diskret. matematika, 19:4 (2007), 23–41 | DOI | MR | Zbl

[28] Dyakonova E., “On subcritical multi-type branching process in random environment”, Algorithms, trees, combinatorics and probabilities, Proc. Fifth Colloquium on Mathematics and Computer Science, DMTCS, Nancy, 2008, 401–408 | MR

[29] Dyakonova E.E., Geiger J., Vatutin V.A., “On the survival probability and a functional limit theorem for branching processes in random environment”, Markov Processes Relat. Fields, 10 (2004), 289–306 | MR | Zbl

[30] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 2, Mir, M., 1984

[31] Fleischmann K., Siegmund-Schultze R., “The structure of reduced critical Galton–Watson processes”, Math. Nachr., 79 (1977), 233–241 | DOI | MR

[32] Fleischmann K., Vatutin V.A., “Reduced subcritical Galton–Watson processes in a random environment”, Adv. Appl. Probab., 31 (1999), 88–111 | DOI | MR | Zbl

[33] Geiger J., Kersting G., “The survival probability of a critical branching process in random environment”, Teoriya veroyatn. i ee primen., 45:3 (2000), 607–615 | DOI | MR | Zbl

[34] Geiger J., Kersting G., Vatutin V.A., “Limit theorems for subcritical branching processes in random environment”, Ann. Inst. Henri Poincaré. Probab. Stat., 39 (2003), 593–620 | DOI | MR | Zbl

[35] Guivarc'h Y., Liu Q., “Propriétés asymptotiques des processus de branchement en environnement aléatoire”, C. r. Acad. sci. Paris. Ser. 1: Math., 332 (2001), 339–344 | DOI | MR | Zbl

[36] Kaplan N., “Some results about multidimensional branching processes with random environments”, Ann. Probab., 2 (1974), 441–455 | DOI | MR | Zbl

[37] Kesten H., Spitzer F., “Convergence in distribution of products of random matrices”, Z. Wahrscheinlichkeitstheor. verwandte Geb., 67 (1984), 363–386 | DOI | MR | Zbl

[38] Kozlov M.V., “Ob asimptotike veroyatnosti nevyrozhdeniya kriticheskikh vetvyaschikhsya protsessov v sluchainoi srede”, Teoriya veroyatn. i ee primen., 21:4 (1976), 813–825 | MR | Zbl

[39] Kozlov M.V., “Uslovnaya funktsionalnaya predelnaya teorema dlya kriticheskogo vetvyaschego protsessa v sluchainoi srede”, DAN, 344:1 (1995), 12–15 | MR | Zbl

[40] Liu Q., “On the survival probability of a branching process in a random environment”, Ann. Inst. Henri Poincaré. Probab. Stat., 32 (1996), 1–10 | MR | Zbl

[41] Sevastyanov B.A., Vetvyaschiesya protsessy, Nauka, M., 1971 | MR | Zbl

[42] Spitser F., Printsipy sluchainogo bluzhdaniya, Mir, M., 1969

[43] Smith W.L., Wilkinson W.E., “On branching processes in random environments”, Ann. Math. Stat., 40 (1969), 814–827 | DOI | MR | Zbl

[44] Tanny D., “Limit theorems for branching processes in a random environment”, Ann. Probab., 5 (1977), 100–116 | DOI | MR | Zbl

[45] Tanny D., “On multitype branching processes in a random environment”, Adv. Appl. Probab., 13 (1981), 464–497 | DOI | MR | Zbl

[46] Vatutin V.A., “Redutsirovannye vetvyaschiesya protsessy v sluchainoi srede: kriticheskii sluchai”, Teoriya veroyatn. i ee primen., 47:1 (2002), 21–38 | DOI | MR | Zbl

[47] Vatutin V.A., “Predelnaya teorema dlya promezhutochno dokriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Teoriya veroyatn. i ee primen., 48:3 (2003), 453–465 | DOI | MR | Zbl

[48] Vatutin V.A., Dyakonova E.E., “Kriticheskie vetvyaschiesya protsessy v sluchainoi srede: veroyatnosti vyrozhdeniya v fiksirovannyi moment”, Diskret. matematika, 9:4 (1997), 100–126 | DOI | MR | Zbl

[49] Vatutin V.A., Dyakonova E.E., “Reduced branching processes in random environment”, Mathematics and computer science. II: Algorithms, trees, combinatorics and probabilities, eds. B. Chauvin, P. Flajolet, D. Gardy, A. Mokkadem, Birkhäuser, Basel, 2002, 455–467 | MR | Zbl

[50] Vatutin V.A., Dyakonova E.E., “Vetvyaschiesya protsessy Galtona–Vatsona v sluchainoi srede. I: Predelnye teoremy”, Teoriya veroyatn. i ee primen., 48:2 (2003), 274–300 | DOI | MR | Zbl

[51] Vatutin V.A., Dyakonova E.E., “Vetvyaschiesya protsessy Galtona–Vatsona v sluchainoi srede. II: Konechnomernye raspredeleniya”, Teoriya veroyatn. i ee primen., 49:2 (2004), 231–268 | DOI | MR | Zbl

[52] Vatutin V., Dyakonova E., “Yaglom type limit theorem for branching processes in random environment”, Mathematics and computer science. III: Algoritms, trees, combinatorics and probabilities, eds. M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger, Birkhäuser, Basel, 2004, 375–385 | MR | Zbl

[53] Vatutin V.A., Dyakonova E.E., “Vetvyaschiesya protsessy v sluchainoi srede i butylochnye gorlyshki v evolyutsii populyatsii”, Teoriya veroyatn. i ee primen., 51:1 (2006), 22–46 | DOI | MR

[54] Vatutin V.A., Dyakonova E.E., “Predelnye teoremy dlya redutsirovannykh vetvyaschikhsya protsessov v sluchainoi srede”, Teoriya veroyatn. i ee primen., 52:2 (2007), 271–300 | DOI | MR

[55] Vatutin V.A., Dyakonova E.E., “Volny v redutsirovannykh vetvyaschikhsya protsessakh v sluchainoi srede”, Teoriya veroyatn. i ee primen., 53:4 (2008), 665–683 | DOI | MR

[56] Vatutin V.A., Dyakonova E.E., “Asimptoticheskie svoistva mnogotipnykh kriticheskikh vetvyaschikhsya protsessov, evolyutsioniruyuschikh v sluchainoi srede”, Diskret. matematika, 22:2 (2010), 22–40 | DOI | MR | Zbl

[57] Vatutin V.A., Kyprianou A.E., “Branching processes in random environment die slowly”, Algorithms, trees, combinatorics and probabilities, Proc. Fifth Colloquium on Mathematics and Computer Science, DMTCS, Nancy, 2008, 379–400 | MR

[58] Vatutin V.A., Vakhtel V.I., “Vnezapnoe vyrozhdenie kriticheskogo vetvyaschegosya protsessa v sluchainoi srede”, Teoriya veroyatn. i ee primen., 54:3 (2009), 417–438 | DOI | MR | Zbl

[59] Vatutin V.A., Zubkov A.M., “Branching processes. II”, J. Sov. Math., 67:6 (1993), 3407–3485 | DOI | MR | Zbl

[60] Weissner E.W., “Multitype branching processes in random environments”, J. Appl. Probab., 8 (1971), 17–31 | DOI | MR | Zbl

[61] Zubkov A.M., “Predelnye raspredeleniya rasstoyaniya do blizhaishego obschego predka”, Teoriya veroyatn. i ee primen., 20:3 (1975), 614–623 | MR | Zbl