Geodesic
Asymptotic properties of multitype critical branching processes evolving in a~random environment
Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 22-40.

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For an extended class of multitype critical branching processes in a random environment, the asymptotic behaviour of the survival probability is found under the conditions which are weaker than those known earlier even for the single-type case. A functional limit theorem is proved for the number of particles in the process at moments $nt$, $0\leq t\leq1$, conditioned on its survival up to moment $n$. 
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V. A. Vatutin; E. E. Dyakonova. Asymptotic properties of multitype critical branching processes evolving in a~random environment. Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 22-40. http://geodesic.mathdoc.fr/item/DM_2010_22_2_a1/

[1] Smith W. L., Wilkinson W., “On branching processes in random environment”, Ann. Math. Statist., 40:3 (1969), 814–827 | DOI | MR | Zbl

[2] Athreya K. B., Karlin S., “On branching processes with random environments: I”, Ann. Math. Statist., 42:5–6 (1971), 1499–1520 | DOI | MR | Zbl

[3] Tanny D., “On multitype branching processes in a random environment”, Adv. Appl. Probab., 13:3 (1981), 464–497 | DOI | MR | Zbl

[4] Weissener E. W., “Multitype branching processes in random environments”, J. Appl. Probab., 8:1 (1971), 17–31 | DOI | MR

[5] Kaplan N., “Some results about multidimensional branching processes with random environments”, Ann. Probab., 2:3 (1974), 441–455 | DOI | MR | Zbl

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

[7] Afanasyev V. I., Geiger J., Kersting G., Vatutin V. A., “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673 | DOI | MR | Zbl

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

[9] Vatutin V. A., Dyakonova E. E., “Vetvyaschiesya protsessy Galtona–Vatsona v sluchainoi srede, I: predelnye teoremy”, Teoriya veroyatnostei i ee primeneniya, 48:2 (2003), 274–300 | MR | Zbl

[10] Vatutin V. A., Dyakonova E. E., “Vetvyaschiesya protsessy Galtona–Vatsona v sluchainoi srede, II: konechnomernye raspredeleniya”, Teoriya veroyatnostei i ee primeneniya, 49:2 (2004), 231–268 | MR | Zbl

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

[12] Vatutin V. A., Kyprianou A. E., “Branching processes in random environment die slowly”, Proc. DMTCS, AI, 2008, 375–396 | MR

[13] Vatutin V. A., Topchii V. A., “Maksimum kriticheskikh protsessov Galtona–Vatsona i nepreryvnye sleva sluchainye bluzhdaniya”, Teoriya veroyatnostei i ee primeneniya, 42:1 (1997), 21–34 | MR | Zbl

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

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

[16] Dyakonova E. E., “Kriticheskie mnogotipnye vetvyaschiesya protsessy v sluchainoi srede”, Diskretnaya matematika, 19:4 (2007), 23–41 | MR | Zbl

[17] Dyakonova E. E., Geiger J., Vatutin V. A., “On the survival probability and a functional limit theorem for branching processes in random environment”, Markov Process. Related Fields, 10:2 (2004), 289–306 | MR | Zbl

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

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

[20] von Bahr B., Esseen C.-G., “Inequalities for the $r$th moment of a sum of random variables, $1\leq r\leq2$”, Ann. Math. Statist., 3 (1965), 299–303

[21] Bertoin J., Doney R. A., “On conditioning a random walk to stay nonnegative”, Ann. Probab., 22 (1994), 2152–2167 | DOI | MR | Zbl

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

[23] Rogozin B. A., “Raspredelenie pervogo lestnichnogo momenta i vysoty i fluktuatsii sluchainogo bluzhdaniya”, Teoriya veroyatnostei i ee primeneniya, 16:4 (1971), 593–613 | MR | Zbl

[24] Tanaka H., “Time reversal of random walks in one dimension”, Tokyo J. Math., 12 (1989), 159–174 | DOI | MR | Zbl

[25] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, Moskva, 1984 | Zbl