Statistical Independence in Probability, Analysis, and Number Theory

Abstract

Professor Kac's monograph is designed to illustrate how simple observations can be made the starting point of rich and fruitful theories and how the same theme recurs in seemingly unrelated disciplines. An elementary but thorough discussion of the game of 'heads or tails', including the normal law and the laws of large numbers, is presented in a setting in which a variety of purely analytic results appear natural and inevitable. The chapter 'Primes Play a Game of Chance' uses the same setting in dealing with problems of the distribution of values of arithmetic functions. The final chapter 'From Kinetic Theory to Continued Fractions' deals with a spectacular application of the ergodic theorems to continued fractions. Mark Kac conveyed his infectious enthusiasm for mathematics and its applications in his lectures, papers, and books. Two of his papers won Chauvenet awards for expository excellence. Born in Poland, he studied with Hugo Steinhaus at Lwów earning his doctorate in 1936. He had a long and productive career in the United States, serving on the faculties of Cornell University (1939-1961), Rockefeller University (1961-1982), and the University of Southern California (1982 until his death in 1984).