HOME

TheInfoList



OR:

Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, specializing in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
theory. The theory of martingales was developed by Doob.


Early life and education

Doob was born in
Cincinnati, Ohio Cincinnati ( ) is a city in the U.S. state of Ohio and the county seat of Hamilton County. Settled in 1788, the city is located at the northern side of the confluence of the Licking and Ohio rivers, the latter of which marks the state line wi ...
, February 27, 1910, the son of a Jewish couple, Leo Doob and Mollie Doerfler Doob. The family moved to
New York City New York, often called New York City or NYC, is the most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the most densely populated major city in the Un ...
before he was three years old. The parents felt that he was underachieving in grade school and placed him in the
Ethical Culture School Ethical Culture Fieldston School (ECFS), also referred to as Fieldston, is a private independent school in New York City. The school is a member of the Ivy Preparatory School League. The school serves approximately 1,700 students with 480 facul ...
, 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 (''Boundary Values of Analytic Functions'', advisor Joseph L. Walsh) in 1932. After postdoctoral research at Columbia and
Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
, he joined the Department of Mathematics of the
University of Illinois The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univer ...
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 from 1942 to 1945; he was at the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
for the academic year 1941–1942 when
Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lon ...
approached him to work on mine warfare for the Navy.


Work

Doob's thesis was on boundary values of analytic functions. He published two papers based on this thesis, which appeared in 1932 and 1933 in the Transactions of the American Mathematical Society. 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 University 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 Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
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. 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, in which he proved theorems related to the law of large numbers, using 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 In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statis ...
for estimating a parameter of a distribution. After writing a series of papers on the foundations of probability and stochastic processes including martingales,
Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
es, and
stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
es, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes, so he wrote the book ''Stochastic Processes''. It was published in 1953 and soon became one of the most influential books in the development of modern probability theory. Beyond this book, Doob is best known for his work on martingales and probabilistic
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
. After he retired, Doob wrote a book of over 800 pages: ''Classical Potential Theory and Its Probabilistic Counterpart''. The first half of this book deals with classical potential theory and the second half with
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, especially martingale theory. In writing this book, Doob shows that his two favorite subjects, martingales and potential theory, can be studied by the same mathematical tools. The
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
's Joseph L. Doob Prize, endowed in 2005 and awarded every three years for an outstanding mathematical book, is named in Doob's honor.


Honors

* President of the
Institute of Mathematical Statistics The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts o ...
in 1950. * Elected to National Academy of Sciences 1957. * President of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
1963–1964. * Elected to
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
1965. * Associate of the French Academy of Sciences 1975. * Awarded the
National Medal of Science The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social scienc ...
by the
President of the United States The president of the United States (POTUS) is the head of state and head of government of the United States of America. The president directs the executive branch of the federal government and is the commander-in-chief of the United States ...
Jimmy Carter James Earl Carter Jr. (born October 1, 1924) is an American politician who served as the 39th president of the United States from 1977 to 1981. A member of the Democratic Party, he previously served as the 76th governor of Georgia from 1 ...
1979. * Awarded the Steele Prize by the American Mathematical Society. 1984.


Publications

;Books * * * ;Articles * * * * * * * * * *


See also

*
Martingale (probability theory) In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
* Doob–Dynkin lemma * Doob martingale *
Doob's martingale convergence theorems In mathematicsspecifically, in the stochastic processes, theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limit (mathematics), limits of Martingale (probability theory), supermartingales, named ...
* Doob's martingale inequality *
Doob–Meyer decomposition theorem The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for ...
* Optional stopping theorem


Notes


External links

*
A Conversation with Joe Doob


{{DEFAULTSORT:Doob, Joseph Leo 1910 births 2004 deaths 20th-century American mathematicians 21st-century American mathematicians Columbia University staff Harvard University alumni Institute for Advanced Study visiting scholars Jewish American scientists Mathematical analysts Mathematicians from Ohio Members of the French Academy of Sciences National Medal of Science laureates People from Cincinnati Presidents of the American Mathematical Society Presidents of the Institute of Mathematical Statistics Princeton University staff Probability theorists University of Illinois Urbana-Champaign faculty Mathematical statisticians