Petrov, V. V. Limit theorems for the sums of independent random variables. (Предельные теоремы для сумм независимых случайных величин.) (Russian) Zbl 0621.60022 Teoriya Veroyatnosteĭ i Matematicheskaya Statistika, Vyp. 39. Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 320 p. R. 3.30 (1987). This monograph is the second edition of the author’s book ”Sums of independent random variables.” Moscow. Nauka (1972; Zbl 0267.60055). This second edition is supplemented with new rich material adopted from original researches concerning limit theorems for sums of independent random variables and related fields conducted in many countries after the first edition appeared. On the other hand some topics of the previous edition are omitted, in particular, the chapter dealing with local limit theorems. The author is an actively working mathematician and one can find his own results practically in all chapters. We shall not dwell at length on them and only mention his profound results on the law of large numbers and the law of iterated logarithm, presented in the last two chapters. Not only some original results included in the book are due to the author, but also a great number of methodical innovations. It will not be an exaggeration to say that the author’s book contains all the most significant results in the theory of summation of independent random variables obtained by this time. His new monograph will undoubtedly win prestige and popularity among the specialists and students who desire to take part in the research work in the field of probability theory. Now we shall briefly dwell on the contents of the book. The first chapter is auxiliary and contains the basic concepts of probability theory and some statements supposed known. The second chapter contains information on infinitely divisible distributions. Besides standard facts one can find here an interesting Esséen’s inequality which subsequently plays an important role. This inequality is also known as one of possible estimates of a concentration function of infinitely divisible distributions. The supplement contains various information on analytical properties of infinitely divisible distributions. A great number of inequalities for distributions of sums of independent random variables are contained in the third chapter. In particular, one can find here inequalities for concentration functions. Many known inequalities are given in a more exact form as a rule due to the author. The supplement contains a great set of inequalities for generalized moments, in particular, for exponential moments. The fourth chapter is devoted to the problems connected with the convergence of distributions of sums of independent random variables to infinitely divisible ones. The way the material given traditional and resembles that in ”Probability theory” of M. Loève [Princeton etc.: Van Nostrand (1963; Zbl 0108.14202)] and in ”Limit distributions for sums of independent random variables” by B. V. Gnedenko and A. N. Kolmogorov [Cambridge: Addison-Wesley (1954; Zbl 0056.36001)]. On the other hand, the supplement contains many new statements. In particular, the theorems stating necessary and sufficient conditions for the convergence of sums of independent random variables to normal distributions when the summands are not supposed to fit the condition of uniform negligibity and for the convergence of moments of sums to the moments of the limit distribution can be found here. The fifth chapter deals with the rate of convergence in the central limit theorem. Parallel with the classical results of A. C. Berry and C.-G. Esseen, new results due to the author himself, L. V. Osipov, C. C. Heyde, P. Hall and others are contained in this chapter; particularly, Osipov’s theorems on the asymptotic expansions of the distribution functions of sums of independent random variables. The supplement contains a great number of statements concerning the rate of convergence in the central limit theorem expressed in terms of pseudomoments and other non- traditional characteristics. Laws of large numbers, convergence of series and the order of growth of the sums of independent random variables are discussed in the sixth chapter. Most of the conditions presented in the statements are necessary and sufficient. The author manages to achieve this by introducing special classes of functions in terms of which the results concerning the order of growth are formulated. The supplement contains further results connected with these laws, the rate of convergence in them and the conditions under which these laws hold expressed in terms of the summands themselves. The last chapter is devoted to the law of iterated logarithm. Besides the classical theorems of Kolmogorov and Hartman-Wintner, some theorems due to the author, in particular his theorem on the interconnection between the rate of convergence in the central limit theorem and the validity of the law of iterated logarithm, are presented here. One can also find here the generalized iterated logarithm law due to the author and A. I. Martikainen. Among other results the supplement contains some theorems stating necessary and sufficient conditions for the law of iterated logarithm. A large list of references to papers connected with the problems mentioned in the book completes the monograph. Reviewer: V. M. Kruglov (Moskva) Cited in 3 ReviewsCited in 114 Documents MSC: 60Fxx Limit theorems in probability theory 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60G50 Sums of independent random variables; random walks Keywords:law of large numbers; law of iterated logarithm; Esseen’s inequality; infinitely divisible distributions; inequalities for generalized moments; necessary and sufficient conditions for the convergence of sums; rate of convergence; central limit theorem; necessary and sufficient conditions for the law of iterated logarithm Citations:Zbl 0267.60055; Zbl 0108.14202; Zbl 0056.36001 × Cite Format Result Cite Review PDF