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A fixed point theorem and its application to the central limit theorem. (English) Zbl 0553.60028

At first, the authors introduce a metric on a space of distribution functions and then define a self-mapping of this space which is proved to be contractive. Subsequently, the authors use their fixed point result to give a proof of the central limit theorems for sequences of sub- independent identically distributed random variables.
A similar idea was used by H. F. Trotter [Arch. Math. 10, 226-234 (1959; Zbl 0086.340)] to give a proof of the central limit theorem. Although he gives a proof in which the use of characteristic functions is avoided, this will limit his result to the case of independent random variables. The approach in this paper is considerably different and the result requires only the weaker assumption of sub-independence.
Reviewer: K.Chung

MSC:

60F05 Central limit and other weak theorems
60E05 Probability distributions: general theory

Citations:

Zbl 0086.340
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References:

[1] H.Cram?r, Mathematical methods of statistics. Princeton U. Press 1964.
[2] R. R. Goldberg, Methods of real analysis. Blaisdell, New York 1964. · Zbl 0138.03501
[3] M.Lo?ve, Probability theory (3rd ed.). Princeton, N. J. 1963.
[4] J. F. Trotter, An elementary proof of the central limit theorem. Arch. Math.10, 226-234 (1959). · Zbl 0086.34002 · doi:10.1007/BF01240790
[5] S.Wilks, Mathematical statistics. New York 1963. · Zbl 0060.29502
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