Abstract
We study the logarithmic average of Sinai's one-dimensional random walk in random environment.
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REFERENCES
I. Berkes, Results and problems related to the pointwise central limit theorem, Asymptotic methods in probability and statistics (Ottawa, ON, 1997) (A volume in honour of Miklós Csörgő, edited by B. Szyszkowicz), pp. 59–96, North-Holland, Amsterdam, 1998.
I. Berkes, E. Csáki and L. Horváth, Almost sure central limit theorems under minimal conditions, Statist. Probab. Lett. 37 (1998), 67–76.
G. A. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561–574.
T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab. 14 (1986) 1206–1218.
E. Csáki and A. Földes, On the logarithmic average of additive functionals, Statist. Probab. Lett. 22 (1995), 261–268.
E. Csáki and A. Földes, On two ergodic properties of self-similar processes, Asymptotic methods in probability and statistics (Ottawa, ON, 1997) (A volume in honour of Miklós Csörgő, edited by B. Szyszkowicz), pp. 97–111, North-Holland, Amsterdam, 1998.
E. Csáki, A. Földes and P. Rèvèsz, On almost sure local and global central limit theorems, Probab. Theory Related Fields 97 (1993), 321–337.
M. Csörgő and L. Horváth, Invariance principles for logarithmic averages, Math. Proc. Cambridge Philos. Soc. 112 (1992), 195–205.
A. Fisher, Convex-invariant means and a pathwise central limit theorem, Adv. in Math. 63 (1987), 213–246.
A. O. Golosov, On limiting distributions for a random walk in a critical onedimensional random environment, Russian Math. Surveys 41 (1986), 199–200.
Y. Hu, Tightness and return time in random environment, Stoch. Proc. Appl. 86 (2000), 81–101.
Y. Hu and Z. Shi, The limits of Sinai's simple random walk in random environment, Ann. Probab. 26 (1998), 1477–1521.
B. D. Hughes, Random Walks and Random Environments. Vol. II: Random Environments, Oxford Science Publications, Oxford, 1996.
I. Ibragimov and M. Lifshits, On the convergence of generalized moments in almost sure central limit theorem, Statist. Probab. Lett. 40 (1998), 343–351.
M. T. Lacey and W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990) 201–205.
K. Kawazu, Y. Tamura and H. Tanaka, Limit theorems for one-dimensional diffusions and random walks in random environments, Probab. Th. Rel. Fields 80 (1989), 501–541.
J. Kesten, The limit distribution of Sinai's random walk in random environment, Physica 138A (1986), 299–309.
J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent R.V.'s and the sample DF. I, Z. Wahrscheinlichkeitstheorie verw. Gebiete 32 (1975), 111–131.
P. Révész, Random Walk in Random and Non-Random Environments, World Scienfitic, Singapore, 1990.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 2nd Edition. Springer, Berlin, 1994.
P. Schatte, On the central limit theorem with almost sure convergence, Probab. Math. Statist. 11 (1990), 237–246.
S. Schumacher, Diffusions with random coefficients, Contemp. Math. 41 (1985), 351–356.
Z. Shi, A local time curiosity in random environment, Stoch. Proc. Appl. 76 (1998), 231–250.
Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium (English translation), Th. Probab. Appl. 27 (1982), 256–268.
F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1–31.
H. Tanaka, Diffusion processes in random environments, Proc. ICM (S. D. Chatterji, ed.) pp. 1047–1054, Birkhäuser, Basel, 1995.
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Hu, Y. The Logarithmic Average of Sinai's Walk in Random Environment. Periodica Mathematica Hungarica 41, 175–185 (2000). https://doi.org/10.1023/A:1010372522910
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DOI: https://doi.org/10.1023/A:1010372522910