Waves in Reduced Branching Processes in a Random Environment

V. A. Vatutin   ; E. E. D'yakonova

Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 4, pp. 665-683

Résumé

Let $Z(n)$, $n=0,1\dots,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),\dots,$ and let $S_{0}=0$, $S_{k}=X_{1}+\dots+X_{k}$, $k\ge1,$ be the associated random walk with $X_{i}=\log f_{i-1}'(1),$ and $\tau (n)$ be the leftmost point of the minimum of $\{ S_{k}$,$k\ge0\} $ on the interval $[0,n]$. Denoting by $Z(k,m)$ the number of particles existing in the branching process at the time moment $k\le m$ which have nonempty offspring at the time moment $m$, and assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0)\to \rho \in (0,1)$, $n\to\infty$, we prove (under the quenched approach) conditional limit theorems, as $n\to\infty$, for the distribution of $Z(nt_{1},nt_{2})$, $0<t_{1}<t_{2}<1,$ given $Z(n)>0$. It is shown that the form of the limit distributions essentially depends on the position of $\tau (n)$ with respect to the interval $[nt_{1},nt_{2}].$

Détail

Keywords: branching processes in a random environment, Doney condition, conditional limit theorems.

Bibliographie

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

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

[3] Athreya K. B., Ney P. E., Branching Processes, Springer-Verlag, New York, Heidelberg, 1972, 287 pp. | MR | Zbl

[4] 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

[5] Borovkov K. A., Vatutin V. A., “Reduced critical branching processes in random environment”, Stochastic Process. Appl., 71:2 (1997), 225–240 | DOI | MR | Zbl

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

[7] Vatutin V. A., Dyakonova E. E., “Reduced branching processes in random environment”, Mathematics and Computer Science, v. II, Algorithms, Trees, Combinatorics and Probabilities, eds. B. Chauvin et al., Birkhäuser, Basel, 2002, 455–467 | MR | Zbl

[8] Vatutin V. A., Dyakonova E. E., “Yaglom type limit theorem for branching processes in random environment”, Mathematics and Computer Science, v. III, Algorithms, Trees, Combinatorics, and Probabilities, eds. M. Drmota et al., Birkhäuser, Basel, 2004, 375–385 | MR | Zbl

[9] 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 | MR | Zbl

[10] Vatutin V. A., Dyakonova E. E., “Vetvyaschiesya protsessy Galtona–Vatsona v sluchainoi srede. II: konechnomernye raspredeleniya”, Teoriya veroyatn. i ee primenen., 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 veroyatn. i ee primen., 51:1 (2006), 22–46 | MR

[12] 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 | MR

[13] Doney R. A., “Spitzer's condition and the ladder variables in random walks”, Probab. Theory Related Fields, 101:4 (1995), 577–580 | DOI | MR | Zbl

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

[15] Spitser F., Printsipy sluchainogo bluzhdaniya, Mir, M., 1969, 472 pp.

[16] Fleischmann K., Vatutin V. A., “Reduced subcritical Galton–Watson processes in a random environment”, Adv. Appl. Probab., 31:1 (1999), 88–111 | DOI | MR | Zbl

[17] Fleischmann K., Prehn U., “Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen”, Math. Nachr., 64 (1974), 357–362 | DOI | MR | Zbl

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