Introduction to Probability

CHAPTER 3INDEPENDENCE AND MULTIVARIATE DISTRIBUTIONS

Yuri Linnik (Bila Tserkva 1915 – Leningrad 1972)

Yuri Vladimirovich Linnik was born on 8 January 1915, in Bila Tserkva (in present‐day Ukraine) and died on 30 June 1972, in Leningrad (in present‐day St. Petersburg, Russia). He studied in St. Petersburg University under the supervision of Professor Vladimir Tartakovski, and subsequently worked in that University and in the Steklov Institute.

He is known for Linnik's large sieve, Linnik's theorem in analytic number theory, and Linnik's ergodic method. He developed many fine results on infinite divisible distributions, which resulted in the book Decomposition of Random Variables and Vectors, collaboratively with I.V. Ostrovskii. He also established numerous characterization results for statistical distributions, many of which are based on independence of certain statistics, and these resulted in the book Characterization Problems in Mathematical Statistics, jointly with A.M. Kagan and C.R. Rao.

For his fine work and accomplishments, he received both the State Prize and the Lenin Prize of the Soviet Union.

3.1 INTRODUCTION

The concept of stochastic independence of random variables is one of the most important concepts in probability theory. In the case when two random variables are independent, knowledge of the distribution for each of them is sufficient to determine their ...