Intermediately Subcritical Branching Process in a Random Environment: The Initial Stage of the Evolution
Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 129-144.

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We consider a branching process evolving in an i.i.d. random environment. It is assumed that the process is intermediately subcritical. We investigate the initial stage of the evolution of the process given its survival for a long time.
Keywords: branching process, random environment, random walk, change of measure.
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E. E. Dyakonova. Intermediately Subcritical Branching Process in a Random Environment: The Initial Stage of the Evolution. Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 129-144. http://geodesic.mathdoc.fr/item/TRSPY_2022_316_a9/

[1] Afanasyev V.I., Böinghoff Ch., Kersting G., Vatutin V.A., “Conditional limit theorems for intermediately subcritical branching processes in random environment”, Ann. Inst. Henri Poincaré. Probab. stat., 50:2 (2014), 602–627 | MR | Zbl

[2] 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

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

[4] Athreya K.B., Karlin S., “Branching processes with random environments. II: Limit theorems”, Ann. Math. Stat., 42:6 (1971), 1843–1858 | DOI | MR | Zbl

[5] Bertoin J., Lévy processes, Cambridge Tracts Math., 121, Cambridge Univ. Press, Cambridge, 1996 | MR

[6] Bertoin J., Doney R.A., “On conditioning a random walk to stay nonnegative”, Ann. Probab., 22:4 (1994), 2152–2167 | DOI | MR | Zbl

[7] Bingham N.H., Goldie C.M., Teugels J.L., Regular variation, Encycl. Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[8] Chaumont L., “Excursion normalisée, méandre et pont pour les processus de Lévy stables”, Bull. sci. math., 121:5 (1997), 377–403 | MR | Zbl

[9] Chauvin B., Rouault A., Wakolbinger A., “Growing conditioned trees”, Stoch. Process. Appl., 39:1 (1991), 117–130 | DOI | MR | Zbl

[10] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, J. Wiley Sons, New York, 1971 | MR | MR | Zbl

[11] Kallenberg O., “Stability of critical cluster fields”, Math. Nachr., 77 (1977), 7–43 | DOI | MR | Zbl

[12] Kersting G., Vatutin V., Discrete time branching processes in random environment, J. Wiley Sons, Hoboken, NJ, 2017 | Zbl

[13] Lyons R., Pemantle R., Peres Y., “Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes”, Ann. Probab., 23:3 (1995), 1125–1138 | DOI | MR | Zbl

[14] Neveu J., “Erasing a branching tree”, Adv. Apl. Probab., 18 (Suppl.) (1986), 101–108 | MR | Zbl

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

[16] Vatutin V., Dyakonova E., “Path to survival for the critical branching processes in a random environment”, J. Appl. Probab., 54:2 (2017), 588–602 | DOI | MR | Zbl

[17] V. A. Vatutin and E. E. Dyakonova, “The initial evolution stage of a weakly subcritical branching process in a random environment”, Theory Probab. Appl., 64:4 (2020), 535–552 | DOI | MR | Zbl