Joseph L. Doob

Joseph L. Doob (1969)

Joseph Leo "Joe" Doob (born February 27, 1910 in Cincinnati , Ohio , † June 7, 2004 in Urbana , Illinois ) was an American mathematician who dealt with analysis and probability theory ( stochastic processes ).

life and work

Doob moved with his parents to New York City in 1913 , where he attended school until he graduated in 1926. He then studied at Harvard University , where he in 1931 his master took Accounts (studied under him at William Fogg Osgood ) and 1932 at Joseph L. Walsh with the dissertation Boundary Values of Analytic Functions doctorate was. After stays at Columbia University , where he worked with the analyst Joseph Ritt and then turned to probability theory with the statistician Harold Hotelling , and in Princeton (with a grant from the Carnegie Foundation ), he went in 1935 as an associate professor at the University of Illinois at Urbana-Champaign , where he was professor from 1945 until his retirement in 1978. He has been a member of the Center for Advanced Study there since it was founded . During World War II he advised (as a civilian) the US Navy in Washington, DC and Guam on mine warfare.

Doob was one of the first who, according to Kolmogorov's axiomatic justification of the calculus of probability within the framework of measure theory, applied its methods in order to provide more stringent proofs for theorems of probability theory. In his first probabilistic work Probability and Statistics from 1934, he used a probabilistic interpretation of Birkhoff's individual ergodic theorem to give rigorous evidence for statistical theorems of Fisher and Hotelling. The concept of Martingales in the theory of stochastic processes goes back to him. His influential book Stochastic Processes was published in 1953 (John Wiley). Originally, he wanted to leave out the theory of measure entirely because statisticians had complained about it. As he noted in his foreword, the resulting first version was so very difficult to understand, which made him further remark in his book: "Probability theory is simply a branch of measure theory, with its own focus and its own special field of application." The name "stochastic" , which subsequently became commonly used for random processes, was chosen because he and William Feller were writing books for the Wiley Series in Statistics at the same time and could not agree on the names "chance variable" and "random variable" for random variables. In 1955 he introduced an axiomatization of the potential theory , which became a further research focus of Doob and brought him back in connection with his roots in function theory . In 1984 Springer published his extensive book Classical potential theory and its probabilistic counterpart , in which he describes the connection between potential theory and stochastic processes (Martingales). In 1993 his book Measure Theory was published by Springer-Verlag.

Doob was President of the American Mathematical Society in 1963-64 . In 1979 he received the US National Medal of Science . In 1984 he received the Leroy P. Steele Prize . He was a member of the American Academy of Arts and Sciences (1965), the National Academy of Sciences and an external member of the French Academy of Sciences . In 1950 he was President of the Institute of Mathematical Statistics . For 25 years he was the commissioner of traditional Saturday Hike at the University of Illinois.

The Doob maximum inequality , the Doob decomposition , the Doob-Meyer decomposition , the Doob martingale , Doob's crossing theorem and the Doob-Dynkin lemma are associated with his name.

His PhD students include David Blackwell and Paul Halmos (his first PhD student in 1938). Halmos donated the Joseph L. Doob Prize at the AMS in his honor .

Fonts

• Stochastic Processes. Wiley, 1953
• Classical potential theory and its probabilistic counterpart. Springer, Berlin [a. a.] 1984, ISBN 3-540-90881-1
• Measure Theory. Springer, Berlin [a. a.] 1993, ISBN 3-540-94055-3
• The development of rigor in mathematical probability (1900-1950). In: Jean-Paul Pier (Ed.): Development of mathematics 1900–1950. Birkhäuser, Basel / Boston / Berlin 1994, ISBN 3-7643-2821-5

literature

• Kai Lai Chung : Probability and Doob. In: American Mathematical Monthly. Volume 105, No. 1, January 1998, p. 28.

Web links

• Interview with Snell, Dartmouth College
• John J. O'Connor, Edmund F. Robertson :  Joseph L. Doob. In: MacTutor History of Mathematics archive .

Footnotes

1. ↑ Transactions of the AMS , Volume 36, 1934, p. 759.
2. ↑ He found the concept implicitly in a book by Jean Ville from 1939. Paul Lévy had done important preparatory work in France .
3. ↑ Originally Norbert Wiener was supposed to act as a co-author. Remnants of the collaboration with Wiener were reflected in a chapter on the prediction of stochastic processes (Chapter 12).
4. ^ "Probability theory is simply a branch of measure theory, with its own special emphasis and field of application." (Doob: Stochastic Processes. P. 1)
5. ↑ after the title of a work by Chintschin