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Teor. Veroyatnost. i Primenen., 1982, Volume 27, Issue 2, Pages 247–258 (Mi tvp2342)  

This article is cited in 25 scientific papers (total in 27 papers)

Limit behaviour of one-dimensional random walks in random environments

Ya. G. Sinaî

Moscow

Abstract: We consider the simplest one-dimensional random walks with transitions $x\to x\pm 1$ having the probabilities $1/2\pm \xi(x)$ where $\xi(x)$ are independent random variables with zero mean and $|\xi(x)|\le c<1/2$. Let $x(n)$ be the position of the moving particle after $n$ steps. We show that the limit distribution of $x(n)/\ln^2n$ is concentrated in a random point depending on a concrete realization of $\xi(\cdot)$.

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English version:
Theory of Probability and its Applications, 1983, 27:2, 256–268

Bibliographic databases:

Received: 25.07.1980

Citation: Ya. G. Sinaî, “Limit behaviour of one-dimensional random walks in random environments”, Teor. Veroyatnost. i Primenen., 27:2 (1982), 247–258; Theory Probab. Appl., 27:2 (1983), 256–268

Citation in format AMSBIB
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\by Ya.~G.~Sina{\^\i}
\paper Limit behaviour of one-dimensional random walks in random environments
\jour Teor. Veroyatnost. i Primenen.
\yr 1982
\vol 27
\issue 2
\pages 247--258
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\zmath{https://zbmath.org/?q=an:0505.60086|0497.60065}
\transl
\jour Theory Probab. Appl.
\yr 1983
\vol 27
\issue 2
\pages 256--268
\crossref{https://doi.org/10.1137/1127028}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. O. Golosov, “On the localization of random walk in a random medium”, Russian Math. Surveys, 39:2 (1984), 157–158  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. I. S. Sineva, “Limit distributions for random walks in a one-dimensional random environment”, Russian Math. Surveys, 40:1 (1985), 242–242  mathnet  crossref  mathscinet  zmath
    3. S. M. Kozlov, “The method of averaging and walks in inhomogeneous environments”, Russian Math. Surveys, 40:12 (1985), 73–145  mathnet  crossref  mathscinet  zmath  adsnasa
    4. A. O. Golosov, “On limiting distributions for a random walk in a critical one-dimensional random environment”, Russian Math. Surveys, 41:2 (1986), 199–200  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. R. L. Dobrushin, Yu. M. Sukhov, J. Fritz, “A. N. Kolmogorov – the founder of the theory of reversible Markov processes”, Russian Math. Surveys, 43:6 (1988), 157–182  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. A. A. Butov, E. V. Krichagina, “A functional limit theorem for a symmetric walk in a random environment”, Russian Math. Surveys, 43:2 (1988), 163–164  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. A. A. Butov, “A limit theorem for a birth and death process in a random medium of functional type”, Russian Math. Surveys, 45:3 (1990), 216–217  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. A. V. Letchikov, “Asymptotic properties with probability 1 for one-dimensional random walks in a random environment”, Math. USSR-Sb., 74:2 (1993), 455–473  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. A. V. Letchikov, “Random walks in one-dimensional quasicrystals”, Theoret. and Math. Phys., 88:3 (1991), 908–912  mathnet  crossref  mathscinet  isi
    10. F. S. Dzheparov, V. E. Shestopal, “Random walks in disordered systems with long-range transitions. Asymptotically exactly solvable models”, Theoret. and Math. Phys., 94:3 (1993), 345–357  mathnet  crossref  mathscinet  isi
    11. A. V. Letchikov, “A criterion for linear drift, and the central limit theorem for one-dimensional random walks in a random environment”, Russian Acad. Sci. Sb. Math., 79:1 (1994), 73–92  mathnet  crossref  mathscinet  zmath  isi
    12. S. P. Novikov, L. A. Bunimovich, A. M. Vershik, B. M. Gurevich, E. I. Dinaburg, G. A. Margulis, V. I. Oseledets, S. A. Pirogov, K. M. Khanin, N. N. Chentsova, “Yakov Grigor'evich Sinai (on his sixtieth birthday)”, Russian Math. Surveys, 51:4 (1996), 765–778  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    13. Ya. G. Sinai, “Anomalous transport in quasi-periodic media”, Russian Math. Surveys, 54:1 (1999), 181–208  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    14. V. E. Shestopal, “Multiparameter models for random walks in disordered lattice systems”, Theoret. and Math. Phys., 119:2 (1999), 660–669  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. Phys. Usp., 42:4 (1999), 297–319  mathnet  crossref  crossref  isi
    16. U. A. Rozikov, “Random walks in random environments of metric groups”, Math. Notes, 67:1 (2000), 103–107  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. Proc. Steklov Inst. Math., 228 (2000), 224–233  mathnet  mathscinet  zmath
    18. F. S. Dzheparov, V. E. Shestopal, “Asymptotically Exactly Solvable Models of Processes in Stochastically Homogeneous Disordered Lattice Media”, Theoret. and Math. Phys., 135:1 (2003), 549–565  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. V. A. Vatutin, E. E. D'yakonova, “Branching processes in random environment and “bottlenecks” in evolution of populations”, Theory Probab. Appl., 51:1 (2007), 189–210  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. Fedorov T.Yu., “Primenenie modeli geometricheskogo sluchainogo bluzhdaniya v sluchainoi srede k opisaniyu finansovykh vremennykh ryadov”, Intellektualnye sistemy v proizvodstve, 2011, no. 1, 54–67  elib
    21. Proc. Steklov Inst. Math., 282 (2013), 106–123  mathnet  crossref  crossref  mathscinet  isi
    22. Bray A.J. Majumdar S.N. Schehr G., “Persistence and First-Passage Properties in Nonequilibrium Systems”, Adv. Phys., 62:3 (2013), 225–361  crossref  isi
    23. A. I. Bufetov, B. M. Gurevich, K. M. Khanin, F. Cellarosi, “The Abel Prize award to Ya. G. Sinai”, Russian Math. Surveys, 69:5 (2014), 931–956  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    24. Shi Zh., “Branching Random Walks Summer School of Probability of Saint-Flour XLII-2012 Preface”: Shi, Z, Branching Random Walks: Ecole D'Ete de Probabilites de Saint-Flour XLII - 2012, Lect. Notes Math., 2151, Springer Int Publishing Ag, 2015, VII+  isi
    25. Xu W., Chen W., Liang Y.-J., Weberszpil J., “A Spatial Structural Derivative Model For Ultraslow Diffusion”, Therm. Sci., 21:1 (2017), S121–S127  crossref  isi
    26. V. I. Afanasyev, “Two-boundary problem for a random walk in a random environment”, Theory Probab. Appl., 63:3 (2019), 339–350  mathnet  crossref  crossref  isi  elib
    27. Hong W. Yang H., “Scaling Limit Theorem For Transient Random Walk in Random Environment”, Front. Math. China, 13:5 (2018), 1033–1044  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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