# Main Page/doob

**Joeseph Leonard Doob ** February 27 1910 - June 27 2004.

## Early life and education

Doob was born in Cincinnati, Ohio, February 27, 1910, the son of Leo Doob and Mollie Doerfler Doob. The family moved to New York City before he was three years old. The parents felt that he was "under-achieving" in grade school and placed him in the Ethical Culture School, from which he graduated in 1926. He then went on to Harvard where he received a BA in 1930, an MA in 1931, and a PhD in 1932. After postdoctoral research at Columbia and Princeton, he joined the Department of Mathematics of the University of Illinois in 1935 and served until his retirement in 1978. He was a member of the Urbana campus's Center for Advanced Study from its beginning in 1959. During the Second World War, he worked in Washington, D. C. and Guam as a civilian consultant to the Navy.

## Work

Doob's thesis was on boundary values of analytic functions. He published two papers based on his thesis which appeared in 1932 and 1933 in the Transactions of the AMS. Doob returned to this subject many years later when he proved a probabilistic version of Fatou's boundary limit theorem for harmonic functions.

The Great Depression of 1929 was still going strong in the thirties and Doob could not find a job. B.O. Koopman at Columbia suggested that statistician Harold Hotelling might have a grant that would permit Doob to work with him. Hotelling did, so the Depression led Doob to probability.

In 1933 Kolmogorov provided the first axiomatic foundation for the theory of probability. Thus a subject that had originated from intuitive ideas suggested by real life experiences and studied informally, suddenly became mathematics. Probability theory became measure theory with its own problems and terminology: expected value for integral, random variable for function, event for subset etc. Doob recognized that this would make it possible to give rigorous proofs for existing probability results, and he felt that the tools of measure theory would lead to new probability results.

Doob's approach to probability was evident in his first probability paper: *Probability and Statistics*, Transactions of the AMS, 1934. Doob proved theorems related to the law of large numbers, using the fact that the law of large numbers follows from a probabilistic interpretation of Birkhoff's ergodic theorem. Then he used these theorems to give rigorous proofs of theorems proven by Fisher and Hotelling related to Fisher's maximum likelihood estimator for estimating a parameter of a distribution.

Åfter writin a series of papers on the foundations of probability, Martingales, Markov Processes Doob realized that there was a real nead for a book where one could find what is known about the varlious types of Stochasict Processes. Sohe wrote such a book called "Stochastic Processes."published in 1953. This book is became one of the most influencial books in the development of probability theory.

Beyond this book Doob is best known for his work on Martingales and Probabilistic Potential Theory. After he retired Doob wrote a book of over 800 pages Classical Potential Theory and Its Probabilistic Counterpart. The first half of this book dealt with classical potential theory and the second half with probability especially martingale theory. in the lprocess Doob shows that his two favorite subjects are really pretty much the same.

## Honors

Doob Honors

President of the Institute of Mathematical Statistics in 1950.

President of the American Mathematical Society 1963-1964.

Elected to American Academy of the Arts and Sciences 1965.

Associate of the French Academy of Sciences 1975.

Awarded the National Medal of Science by President Carter 1979.

Awarded the Steele Prize by the American Mathematical Society. 1984