Natural estimates of convergence rate in the central limit theorem
A retrospective analysis of the limit theorems is given. The reader can find a deep discussion of the most known forms of the central limit theorem, in particular, the Lindeberg-Feller theorem and the Zolotarev theorem. A serious attempt is made to answer the extremely difficult question, what is the mechanism of forming of probabilistic regularities and in what ways this mechanism reveals itself. In this connection the author introduces the notion of the natural estimates of the rate of convergence in the central limit theorem. Let \(\Phi\) and \(F_n\) be the standard normal distribution function and the convolution \(F_n =F_{n,1}*\cdots* F_{n,n}\) of distribution functions, respectively. The lower estimate \(\psi_1(\nu')\) and the upper estimate \(\psi_2(\nu'')\), \[ \psi_1(\nu')\leq \sup_x|F_n(x)- \Phi(x)|\leq \psi_2(\nu''), \] from the natural estimates where \(\nu'\) and \(\nu''\) are some functionals defined on the set \(\{F_{n,1},\dots,\) \(F_{n,n}\}\) and \(\psi_1(u)\) and \(\psi_2(u)\), \(n\geq 0\), are some nonnegative functions such that \(\psi_k(0+)= 0\). The case \(\nu'= \nu''\) is the most interesting but it is unrealistic in the general setting. The author indicates a number of examples when natural estimates can be constructed.
scientific article; zbMATH DE number 1289196
scientific article