Decomposition of Random Variables and Vectors

Abstract

In the Foreword to his book of 1960, Linnik described decomposition of (probability) laws as "a field which in relation to mathematics employed lies between the theory of probability and the theory of functions of a complex variable", nowadays one would add "and of several complex variables." This field stands isolated from the mainstream of probability theory and is largely ignored. Thus it behooves us to be specific about its main concepts, problems, and representative results with their dates, in order to point out the evolution of the field before and after the crucial Linnik book which described its state as of 1960. Until recently the central problem of probability theory was, and in large part still is, that of behaviour of sums of independent random variables. The inverse problem of decomposition of random variables, more precisely of their laws, was born-with its concepts, problems, and methods-during its heroic period 1934-1938 thanks to P. Levy, Cramer, Hincin, and Raikov. For twenty years it attracted little attention except mainly during 1947-1951 when Cramer, Levy and Dugué produced various examples and counterexamples and Dugué introduced "ridge functions" which were to play an important role in factorizations of analytic characteristic functions. Thereafter Linnik's deep results, the impact of his book of 1960, and his personal influence attracted to the field a number of bright young mathematicians, especially Ostrovskiï-the joint author of the book under review. It presents the most exhaustive survey there is of the present state of the field.

The Network Origins of Aggregate Fluctuations

“…The following theorem is from Rotar (1975). A detailed treatment of the subject can be found in Chapter 9 of Linnik and Ostrovskiȋ (1977).…”

Independence of linear forms with random coefficients

“…We also need some well-known results of the theory of entire functions. The proof of these theorems may be found in Chapters 1 and 5 of Levin's book (1964), Chapter 8 of Titchmarsh's book (1968), and Chapter 1-3 of the monograph of Linnik and Ostrovskii (1977).…”

On the Smoothed Minimum Error Entropy Criterion

“…Proof: The first part of this property is a direct consequence of the maximum entropy property of the Gaussian distribution. The second part comes from Cramer's decomposition theorem [24], which states that if 1 X and 2 X are independent and their sum 1 2 X X + is Gaussian, then both 1 X and 2 X must also be Gaussian.…”