Introduction to Probability

CHAPTER 8LIMIT THEOREMS

Harald Cramér (Stockholm, Sweden 1893 – Stockholm 1985)

Harald Cramér was born on 25 September 1893, in Stockholm, Sweden, and died on 5 October 1985, at the age of 92, in Stockholm, Sweden. After completing his degree in Mathematics and Chemistry at the University of Stockholm, he finished his PhD in Mathematics in 1917 under the supervision of Professor Marcel Riesz at the University of Stockholm. Following that, he continued working at the University of Stockholm in the Department of Mathematics and produced many fine works in analytic number theory, probability, and probabilistic number theory. Then, in 1929, he was appointed in Stockholm University as a chair professor of Actuarial Mathematics and Mathematical Statistics. His main focus of work then became actuarial science and mathematical statistics, resulting in several well‐known results such as Cramér‐Rao bound, Cramér's decomposition theorem, and Cramér's theorem on large deviations. In 1946, he published his book Mathematical Methods of Statistics, which became a highly influential book. He went on to become President of Stockholm University in 1950, and then Chancellor of the University in 1958.

8.1 INTRODUCTION

In this final chapter, we discuss one of the cornerstones of probability theory, namely, the Central Limit Theorem (CLT). Before that, in Section 8.2, we state two other important ...