The Stein-Tikhomirov method and a nonclassical central limit theorem (Q5916331)

The author presents a criterion for the validity of a nonclassical central limit theorem in terms of characteristic functions. It has been shown that the arguments used in applying the Stein-Tikhomirov method can be considerably simplified. Also the author demonstrates in the course the proof of a nonclassical central limit theorem, which can be called the generalized Lindeberg-Feller theorem. The main result is stated as follows: Theorem. In order that \(\sup_{x}\|F_n(x) - \Phi(x)\|\to 0\) as \(n\to\infty,\) it is necessary and sufficient that for any \(T>0\) the following relation holds: \( \sup_{|t|\leq T}\sum_{j}\|\Delta(f_{nj}(t)) \|\to 0,\) where \(f_{nj}(\cdot)\) is the characteristic function corresponding to the distribution function \(F_{nj}(x).\) The author claims that the present paper is the first one in which the criterion for the convergence of the distribution of the sum \(S_{n}\) to the normal law is stated in terms of the characteristic functions of the summands.