A fixed point theorem and its application to the central limit theorem
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 \textit{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.
metric on a space of distribution functions
scientific article; zbMATH DE number 3881577