Summary
LetS 1,S 2,... be a sequence of sums of i.i.d. random variables. The properties of the logarithmic average
will be studied under some conditions.
Article PDF
Similar content being viewed by others
References
Berkes, I. Dehling, M.: Some limit theorems in log density. Ann. Probab. (to appear)
Brosamler, G.A.: The asymptotic behaviour of certain additive functionals of Brownian motion. Invent. Math.20, 87–96 (1973)
Brosamler, G.A.: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc.104, 561–574 (1988)
Chung, K.L., Erdős, P.: Probability limit theorems assuming only the first moment. Mem. of the AMS6 (1951)
Csáki, E.: On the local time of Wiener process and random walk. Lecture Notes, 7-th International Summer School on Probability Theory and Mathematical Statistics, Varna, Golden Sauds, Bulgaria 16–27 Sept. 1991 pp. 32–59. Science Culture Technology Publishing 1993
Csáki, E., Csörgő, M., Földes, A., Révész, P.: Strong approximations of additive functionals. J. Theor. Probab.5, 679–706 (1992)
Csörgő, M., Horváth, L.: Invariance principles for logarithmic averages. Math. Proc. Camb. Philos. Soc.112, 195–205 (1992)
Erdős, P., Hunt, G.A.: Changes of signs of sums of random variables. Pac. J. Math.3, 673–687 (1953)
Erdős, P., Taylor, S.J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hung.11, 137–162 (1960)
Fisher, A.: Convex-invariant means and a pathwise central limit theorem. Adv. Math.63, 213–246 (1987)
Fisher, A.: A pathwise central limit theorem for random walks. (Preprint)
Lacey, M.: On almost sure noncentral limit theorems. J. Theor. Probab.4, 767–781 (1991)
Lacey, M., Philipp, W.: A note on the almost sure central limit theorem. Stat. Probab. Lett.9, 201–205 (1990)
Lévy, P.: Théorie de l'addition des Variables Aléatoires. Paris: Gauthier Villars 1937
Peligrad, M., Révész, P.: On the almost sure central limit theorem. In: Bellow, A., Jones, R. (eds.) Almost everywhere convergence II., pp. 209–225. New York: Academic Press 1991
Petrov, V.V.: Sums of independent random variables. New York: Springer 1975
Révész, P.: Random walk in random and non-random environments. Singapore: World Scientific 1990
Schatte, P.: On strong versions of the central limit theorem. Math. Nachr.137, 249–256 (1988)
Schatte, P.: On the central limit theorem with almost sure convergence. Probab. Math. Stat.11, 237–246 (1990)
Schatte, P.: Two remarks on the almost sure central limit theorem. Math. Nachr.154, 225–229 (1991)
Weigl, A.: Zwei Sätze über die Belegungszeit beim Random Walk. Diplomarbeit. TU Wien, 1989
Author information
Authors and Affiliations
Additional information
Dedicated to Paul Erdős on the occasion of his 80th birthday
Research supported by Hungarian National Foundation for Scientific Research, Grant No. 1905 and CUNY Research Grant No. 662349
Rights and permissions
About this article
Cite this article
Csáki, E., Földes, A. & Révész, P. On almost sure local and global central limit theorems. Probab. Th. Rel. Fields 97, 321–337 (1993). https://doi.org/10.1007/BF01195069
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01195069