Classical-Type Limit Theorems for Sums of Independent Random Variables

This article presents a number of classical limit theorems for sums of independent random variables and more recent results which are closely related to the classical theorems. It concentrates on three basic subjects: the central limit theorem, the laws of large numbers and the law of the iterated logarithm for sequences of independent real-valued random variables. The author was restricted to an article of small size. Therefore many chapters of the classical theory of summation of independent random variables were omitted, particularly limit theorems with non-normal limit distributions, multidimensional limit theorems and local limit theorems. This article may be regarded as an introduction for the reader who wishes to become acquainted with the classical limit theorems for sums of independent random variables without spending much time. More detailed presentations may be found, for example, in the author’s book (1987) and in its predecessors Csörgö and Révész (1981), Gnedenko and Kolmogorov (1949), Hall (1982), Ibragimov and Linnik (1965), Loève (1960), Petrov (1972, 1995), Révész (1967) and Stout (1974).