Asymptotic properties with probability 1 for one-dimensional random walks in a random environment

A. V. Letchikov

Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 455-473

Résumé

Random walks in a random environment are considered on the set $\mathbf Z$ of integers when the moving particle can go at most $R$ steps to the right and at most $L$ steps to the left in a unit of time. The transition probabilities for such a random walk from a point $x\in\mathbf Z$ are determined by the vector $\mathbf p(x)\in\mathbf R^{R+L+1}$. It is assumed that the sequence $\{\mathbf p(x),\,x\in\mathbf Z\}$ is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results.

Bibliographie

[1] Chernov A. A., “Reduplikatsiya mnogokomponentnoi tsepochki mekhanizmom «molniya»”, Biofizika, 12:2 (1967), 297–301

[2] Temkin D. E., “K teorii bezdiffuzionnogo rosta kristallov”, Kristallografiya, 14:3 (1969), 423–430

[3] Temkin D. E., “Odnomernye sluchainye bluzhdaniya v dvukhkomponentnoi tsepochke”, DAN SSSR, 206:1 (1972), 27–30 | MR | Zbl

[4] Bernasconi J., Schneider W. R., Wyss W., “Diffusion and hopping conductivity in disordered one-dimensional lattice systems”, Zeit. Phys. B, 37 (1980), 175–184 | DOI

[5] Alexander S., Bernasconi J., Schneder W. R., Orbach R., “Excitation dynamics in random one-dimensional system”, Rev. Modern Phys., 53:2 (1981), 175–198 | DOI | MR

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

[7] Sinai Ya. G., “Predelnoe povedenie odnomernogo sluchainogo bluzhdaniya v sluchainoi srede”, Teoriya veroyatnostei i ee primeneniya, 27:2 (1982), 247–258 | MR | Zbl

[8] Deheuvels P., Revesz P., “Simple random walk on the line in random environment”, Prob. Th. Rel. Fields, 72:2 (1986), 215–230 | DOI | MR | Zbl

[9] Key E., “Recurrence and transience criteria for random walk in a random environment”, Ann. Prob., 12:2 (1984), 529–560 | DOI | MR | Zbl

[10] Letchikov A. V., “Predelnaya teorema dlya vozvratnogo sluchainogo bluzhdaniya v sluchainoi srede”, DAN SSSR, 302:1 (1989), 25–28 | MR

[11] Oseledets V. I., “Multiplikativnaya ergodicheskaya teorema. Kharakteristicheskie pokazateli Lyapunova dinamicheskikh sistem”, Tr. MMO, 19 (1968), 179–210 | Zbl

[12] Furstenberg H., “Non-commuting random products”, Trans. Amer. Math. Soc., 108:3 (1964), 377–408 | DOI | MR

[13] Tutubalin V. N., “O predelnykh teoremakh dlya proizvedeniya sluchainykh matrits”, Teoriya veroyatnostei i ee primeneniya, 10:1 (1965), 19–32 | MR | Zbl

[14] Tutubalin V. N., “A central limit theorem for products random matrices and some of its applications”, Symp. Math., 21 (1977), 101–116 | MR | Zbl

[15] Molchanov S. A., Tutubalin V. N., “Lineinaya model gidromagnitnogo dinamo i proizvedeniya sluchainykh matrits”, Teoriya veroyatnostei i ee primeneniya, 29:2 (1984), 234–247 | MR

[16] Philipp W., Stout W., “Almost sure invariance principles for partial sums of weakly dependent random variables”, Mem. Amer. Math. Soc., 161 (1975), 1–140 | MR | Zbl

[17] Csaki E., “On the lower limits of maxima and minima of Wiener process and partial sums”, Zeit. Wahrschein. Verw. Geb., 43:3 (1978), 205–221 | DOI | MR | Zbl

[18] Lyubich Yu. I., Tabachnikov M. I., “Subgarmonicheskie funktsii na orientirovannom grafe”, Sib. matem. zhurn., 10:3 (1969), 600–608 | MR

[19] Shiryaev A. N., Veroyatnost, Nauka, M., 1980 | MR | Zbl