Sums of Independent Random Variables - Petrov, Valentin V. (Springer Berlin Heidelberg, 1975)
The classic "Limit Distributions for sums of Independent Random Variables" by B. V. Gnedenko and A. N. Kolmogorov was published in 1949. Since then the theory of summation of independent variables has developed rapidly. Today a summing-up of the studies in this area, and their results, would require many volumes. The monograph by I. A. Ibragimov and Yu. V. Linnik, "Independent and Stationarily Connected Variables", which appeared in 1965, contains an exposition of the contemporary state of the theory of the summation of independent identically distributed random variables. The present book borders on that of Ibragimov and Linnik, sharing only a few common areas. Its main focus is on sums of independent but not necessarily identically distributed random variables. It nevertheless includes a number of the most recent results relating to sums of independent and identically distributed variables. Together with limit theorems, it presents many probahilistic inequalities for sums of an arbitrary number of independent variables. The last two chapters deal with the laws of large numbers and the law of the iterated logarithm. These questions were not treated in Ibragimov and Linnik; Gnedenko and Kolmogotov deals only with theorems on the weak law of large numbers. Thus this book may be taken as complementary to the book by Ibragimov and Linnik. I do not, however, assume that the reader is familiar with the latter, nor with the monograph by Gnedenko and Kolmogorov, which has long since become a bibliographical rarity