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On a local theorem. (Russian) Zbl 0697.60030
Let \(\xi_ 1,...,\xi_ n\) be i.i.d. random variables having mean 0 and unit variance. Denote \[ P_ n(x,h)=P(x\leq \xi_ 1+...+\xi_ n<x+h). \] In the case when \(\xi_ 1\) has a non-lattice distribution, conditions are given in terms of \(P_ 1(x,h)\) under which there exist \(x_ n\) such that for \(x>x_ n\) and any fixed \(h>0\) \[ P_ n(x,h)=nP_ 1(x,h)(1+o(1))\quad as\quad n\to \infty. \] This is a local version of A. V. Nagaev’s theorem in Teor. Veroyatn. Primen. 22, 335-346 (1977; Zbl 0376.60055); English translation in Theory Probab. Appl. 22(1977), 326-338 (1978).
Reviewer: T.Shervashidze
60F05 Central limit and other weak theorems