On the time of attaining a high level by a transient random walk in a random environment

V. I. Afanasyev

Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 234-267

Résumé

Let a sequence of independent identically distributed pairs of random variables $(p_{i},q_{i}) $, $i\in \mathbf{Z}$, be given, with ${p_{0}+q_{0}=1}$ and $p_{0}>0$, $q_{0}>0$ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in \mathbf{Z}$. This means that under a fixed environment a walking particle located at some moment in a state $i$ jumps either to the state $(i+1) $ with probability $p_{i}$ or to the state $(i-1) $ with probability $q_{i}$. It is assumed that $\mathbf{E}\,\log (p_{0}/q_{0}) <0$, i.e., that the random walk tends with time to $-\infty$. The set of such random walks may be divided into three types according to the value of the quantity $\mathbf{E}\,((p_{0}/q_{0}) \log (p_{0}/q_{0}))$. In the case when the expectation above is zero we prove a limit theorem as $n\to \infty $ for the of time distribution of reaching the level $n$ by the mentioned random walk.

Bibliographie

[1] Kesten H., Kozlov M. V., Spitzer F., “A limit law for random walk in a random environment”, Compositio Mathem., 30:2 (1975), 145–168 | MR | Zbl

[2] Ritter G. A., Random walk in a random environment, critical case, A thesis, Cornell Univ., Cornell, USA, 1976, 73 pp. | MR

[3] Afanasev V. I., “O vremeni dostizheniya vysokogo urovnya sluchainym bluzhdaniem v sluchainoi srede”, Teoriya veroyatn. i ee primen., 57:4 (2012), 625–648 | DOI

[4] Afanasev V. I., “O maksimume nevozvratnogo sluchainogo bluzhdaniya v sluchainoi srede”, Teoriya veroyatn. i ee primen., 35:2 (1990), 209–219 | MR

[5] Solomon F., “Random walks in a random environment”, Ann. Probab., 3:1 (1975), 1–31 | DOI | MR | Zbl

[6] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 738 pp.

[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] Vatutin V. A., Kyprianou A. E., “Branching processes in random environment die slowly”, Discrete Math. Theoret. Comput. Sci., DMTCS, AI, 2008, 375–396 | MR

[9] Boeinghoff Ch., “Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions”, Stoch. Proc. Appl., 124:11 (2014), 3553–3577 | DOI | MR | Zbl

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

[11] Afanasev V. I., “Uslovnaya predelnaya teorema dlya maksimuma sluchainogo bluzhdaniya v sluchainoi srede”, Teoriya veroyatn. i ee primen., 58:4 (2013), 625–647 | DOI

[12] Nagaev S. V., “O skorosti skhodimosti raspredeleniya maksimuma summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 15:2 (1970), 320–325 | MR

[13] Afanasev V. I., “O sootnoshenii maksimalnogo i obschego chisla chastits v kriticheskom vetvyaschemsya protsesse v sluchainoi srede”, Teoriya veroyatn. i ee primen., 48:3 (2003), 435–452 | DOI | Zbl

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

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