Hamedani, G. G.; Walter, G. G. A fixed point theorem and its application to the central limit theorem. (English) Zbl 0553.60028 Arch. Math. 43, 258-264 (1984). 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 Cited in 2 ReviewsCited in 2 Documents MSC: 60F05 Central limit and other weak theorems 60E05 Probability distributions: general theory Keywords:metric on a space of distribution functions; central limit theorems; sub- independence Citations:Zbl 0086.340 PDFBibTeX XMLCite \textit{G. G. Hamedani} and \textit{G. G. Walter}, Arch. Math. 43, 258--264 (1984; Zbl 0553.60028) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.