Arak, T. V.; Zajtsev, A. Yu. Uniform limit theorems for sums of independent random variables. Transl. from Russian. (English) Zbl 0659.60070 Proceedings of the Steklov Institute of Mathematics, 1. Providence, RI: American Mathematical Society (AMS). vii, 222 p. (1988). The authors of this monograph describe interesting and deep results of Arak on the uniform approximation of the distribution of a sum of i.i.d. random variables by an infinitely divisible distribution in the Kolmogorov metric. These results considerably sharpen earlier results of Kolmogorov (who proved an upper bound of the order \(o(n^{-1/5})\) and later the bound \(o(n^{-1/3}))\) and Prokhorov \((o(n^{-1/3}\ln n))\), and give an upper bound of order \(o(n^{-2/3})\) which is also the lower bound for the worst possible choice of the distribution F of the random variable. These deep results are based on a careful study of the implications of having a lower bound on the concentration function of F. This yields information about the concentration of F about the \(\epsilon\)- neighbourhoods of the one-dimensional projection of a higher m- dimensional set, i.e. about neighbourhoods of sets of the type \(u_ 1n_ 1+...+u_ mn_ m\), \(u_ i\in {\mathbb{R}}\), \(n_ j\in \{-1,0,1\}\). It reflects the study of the c.f. \(\hat F\) for all values of t such that \(| \hat F(t)| \geq 1-\epsilon.\) In order to effectively combine results on characteristic functions and order relations on the probabilities involved, the first author invented the so called ‘method of triangle functions’ which combines the virtues of both approaches. This important method is successfully applied in various cases in this monograph to the estimation of closeness of successive sums. Theorem 4.2 and 5.1 [rates \(O(n^{-1/2})\) and \(O(n^{- 1})\) for \(\hat F(t)>-1+\epsilon\), F symmetric] and the approximation of \(F^ n\) by its poissonization [Theorem 5.4, \(o(n^{-1/2})\), F symmetric and Theorem 5.5, \(O(n^{-1})\), \(\hat F\geq 0]\). This monograph is strongly recommended to anyone interested in concentration inequalities and approximations by infinitely divisible laws. Reviewer: F.Götze Cited in 31 Documents MSC: 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems Keywords:infinitely divisible distribution; Kolmogorov metric; concentration function; concentration inequalities Citations:Zbl 0606.60028 × Cite Format Result Cite Review PDF