Hiroshima Mathematical Journal

A central limit theorem of mixed type for a class of 1-dimensional transformations

Hiroshi Ishitani

Full-text: Open access

Article information

Source
Hiroshima Math. J., Volume 16, Number 1 (1986), 161-188.

Dates
First available in Project Euclid: 21 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1206130545

Digital Object Identifier
doi:10.32917/hmj/1206130545

Mathematical Reviews number (MathSciNet)
MR837320

Zentralblatt MATH identifier
0658.60042

Subjects
Primary: 28D05: Measure-preserving transformations
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes

Citation

Ishitani, Hiroshi. A central limit theorem of mixed type for a class of 1-dimensional transformations. Hiroshima Math. J. 16 (1986), no. 1, 161--188. doi:10.32917/hmj/1206130545. https://projecteuclid.org/euclid.hmj/1206130545


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References

  • [1] Dunford, N. and Schwartz, J. T.: Linear Operators, Part I, Interscience, New York, 1957.
  • [2] Fortet, R.: Sur une suite egalement repartie, Studia Math. 9 (1940), 54-69.
  • [3] Gnedenko, B. V. and Kolmogorov, A. N.: Limit distribution for sums of independent random variables, Addison-Wesley, Reading, Massachusetts, 1954.
  • [4] Hofbauer, F. and Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140.
  • [5] Ibragimov, I. A. and Linnik, Yu V.: Independent and stationary sequences of random variables, Walters-Nordhoff, Groningen, 1971.
  • [6] Ionescu-Tulcea, C. and Marinescu, G.: Theorie ergodique pour des classes d'operations non completement continues, Ann. of Math. 52 (1950), 140-147.
  • [7] Ishitani, H.: The central limit theorem for piecewise linear transformations, Publ. RIMS, Kyoto Univ. 11 (1976), 281-296.
  • [8] Ito, Sh. and Takahashi, Y.: Markov subshifts and realizations of /3-transformations, J. Math. Soc. Japan 26 (1974), 33-55.
  • [9] Ito, Sh., Tanaka, S. and Nakada, H.: On unimodal linear transformations and chaos I, Tokyo J. Math. 2 (1979), 221-239.
  • [10] Ito, Sh., Tanaka, S. and Nakada, H.: On unimodal linear transformations and chaos II, ibid, 241-259.
  • [11] Jabtoski, M. and Malczak, J.: The rate of convergence of iterates of the Frobenius- Perron operator for piecewise convex transformation, Bull. Akad. Polon. Sci., Ser. Math. (in press).
  • [12] Jaboski, M. and Malczak, J.: A central limit theorem for piecewise convex mappings of unit interval, Tohoku Math. J. 35 (1983), 173-180.
  • [13] Kac, M.: On distribution of values of sums of the type >f(2kt), Ann. of Math. 47 (1946), 33^9.
  • [14] Keller, G.: Une Theoreme de la limite centrale pour une classe de transformations monotones par morceaux, C. R. Akad. Sci. Paris 291 (1980), 155-158.
  • [15] Lasota, A. and Yorke, J. A.: On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.
  • [16] Lasota, A. and Yorke, J. A.: Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375-384.
  • [17] Li, T. and Yorke, J. A.: Ergodic transformations from an interval into itself, Trans.Amer. Math. Soc. 235 (1978), 182-192.
  • [18] Morita, T.: Random iteration of one-dimensional transformations, (to appear).
  • [19] Nakada, H.: On the invariant measures and the entropies for continued fractions transformation, Keio Math. Rep. 5 (1980), 37-44.
  • [20] Nakada, H.: Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981), 400-426.
  • [21] Nakada, H., Ito, Sh. and Tanaka, S.: On the invariant measure for the transformations associated with some real continued-fractions, Keio Engin. Rep. 30, No. 13 (1977), 159-175.
  • [22] Rousseau-Egele, J.: Un theoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, (preprint).
  • [23] Tanaka, S. and Ito, Sh.: On a family of continued-fraction transformations and their ergodic properties, Tokyo J. Math. 4 (1981), 153-176.
  • [24] Wilkinson, K.: Ergodic properties of a class of piecewise linear transformations, Z. Wahr. 31 (1975), 303-323.
  • [25] Wong, S.: A central limit theorem for piecewise monotonic mappings of the unit interval, Ann. Prob. 7 (1979), 500-514.