Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, 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 (38 ...

and

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

theory. The theory of martingales was developed by Doob.

Early life and education

Doob was born in

Cincinnati, Ohio Cincinnati ( ; colloquially nicknamed Cincy) is a city in Hamilton County, Ohio, United States, and its county seat. Settled in 1788, the city is located on the northern side of the confluence of the Licking River (Kentucky), Licking and Ohio Ri ...

, 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 (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ...

before he was three years old. The parents felt that he was underachieving in grade school and placed him in the

Ethical Culture School The Ethical Culture Fieldston School (ECFS), also known more simply as Fieldston or Ethical Culture, is a private pre-K through twelfth grade coeducational school in New York City with two campuses, in Manhattan and in the Bronx. The school is ...

, from which he graduated in 1926. He then went on to

Harvard Harvard University is a private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher lear ...

where he received a BA in 1930, an MA in 1931, and a PhD (''Boundary Values of Analytic Functions'', advisor

Joseph L. Walsh __NOTOC__ Joseph Leonard Walsh (September 21, 1895 – December 6, 1973) was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szeg� ...

) in 1932. After postdoctoral research at Columbia and

Princeton Princeton University is a private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the Unit ...

, he joined the department of mathematics of the

University of Illinois The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public university, public land-grant university, land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United ...

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) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

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 lo ...

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 The Great Depression was a severe global economic downturn from 1929 to 1939. The period was characterized by high rates of unemployment and poverty, drastic reductions in industrial production and international trade, and widespread bank and ...

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 Harold Hotelling (; September 29, 1895 – December 26, 1973) was an American mathematical statistician and an influential economic theorist, known for Hotelling's law, Hotelling's lemma, and Hotelling's rule in economics, as well as 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 Soviet ...

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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

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 In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...

, using a probabilistic interpretation of

Birkhoff's ergodic theorem Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavi ...

. Then he used these theorems to give rigorous proofs of theorems proven by

Fisher Fisher is an archaic term for a fisherman, revived as gender-neutral. Fisher, Fishers or The Fisher may also refer to: Places Australia * Division of Fisher, an electoral district in the Australian House of Representatives, in Queensland *Elec ...

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 stati ...

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 In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...

es, and

stationary process In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...

es, Doob realized that there was a real need for a book showing what is known about the various types of

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...

es, 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 the two fundamental forces of nature known at the time, namely g ...

. 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 or probability calculus 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 expre ...

, 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. The postdoctoral members of the department of mathematics of the

are named J L Doob Research Assistant Professors.

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 ...

in 1950. * Elected to

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

1957. * President of the

1963–1964. * Elected to

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) 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, and other ...

1965. * Associate of the

French Academy of Sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...

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 science, behavior ...

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. The president directs the Federal government of the United States#Executive branch, executive branch of the Federal government of t ...

Jimmy Carter James Earl Carter Jr. (October 1, 1924December 29, 2024) was an American politician and humanitarian who served as the 39th president of the United States from 1977 to 1981. A member of the Democratic Party (United States), Democratic Party ...

1979. * Awarded the

Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

by the American Mathematical Society. 1984.

Publications

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

Notes

External links