Joseph Leo Doob
Encyclopedia
Joseph Leo Doob was an American mathematician
, specializing in analysis
and probability
theory.
The theory of martingale
s was developed by Doob.
, 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 underachieving 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.
'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
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 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
es, and stationary process
es, Doob realized that there was a real need for a book showing what is known about the various types of stochastic process
es. So he wrote his famous "Stochastic Processes" book. 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
. 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
, 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
's Joseph L. Doob Prize, endowed in 2005 and awarded every three years for an outstanding mathematical book, is named in Doob's honor.
Articles by Doob
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, specializing in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
theory.
The theory of martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
s was developed by Doob.
Early life and education
Doob was born in Cincinnati, OhioCincinnati, Ohio
Cincinnati is a city in the U.S. state of Ohio. Cincinnati is the county seat of Hamilton County. Settled in 1788, the city is located to north of the Ohio River at the Ohio-Kentucky border, near Indiana. The population within city limits is 296,943 according to the 2010 census, making it Ohio's...
, February 27, 1910, the son of Leo Doob and Mollie Doerfler Doob. The family moved to New York City
New York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
before he was three years old. The parents felt that he was underachieving in grade school and placed him in the Ethical Culture School, from which he graduated in 1926. He then went on to Harvard
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
where he received a BA in 1930, an MA in 1931, and a PhD in 1932. After postdoctoral research at Columbia
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
and Princeton
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, he joined the Department of Mathematics of the University of Illinois
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...
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 this 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 FatouPierre Fatou
Pierre Joseph Louis Fatou was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory...
's boundary limit theorem for harmonic functions.
The Great Depression
Great Depression
The Great Depression was a severe worldwide economic depression in the decade preceding World War II. The timing of the Great Depression varied across nations, but in most countries it started in about 1929 and lasted until the late 1930s or early 1940s...
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
Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...
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 Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
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
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
, 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 for estimating a parameter of a distribution.
After writing a series of papers on the foundations of probability and stochastic processes including martingales
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
, Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
es, and stationary process
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
es, Doob realized that there was a real need for a book showing what is known about the various types of stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es. So he wrote his famous "Stochastic Processes" book. 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
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
and probabilistic potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
. 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
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, 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
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
'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 StatisticsInstitute of Mathematical StatisticsThe 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 of the world...
in 1950. - President of the American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
1963-1964. - Elected to American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
1965. - Associate of the French Academy of SciencesFrench Academy of SciencesThe 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 research...
1975. - Awarded the National Medal of ScienceNational Medal of ScienceThe 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 sciences, biology, chemistry, engineering, mathematics and...
by the President of the United StatesPresident of the United StatesThe President of the United States of America is the head of state and head of government of the United States. The president leads the executive branch of the federal government and is the commander-in-chief of the United States Armed Forces....
Jimmy CarterJimmy CarterJames Earl "Jimmy" Carter, Jr. is an American politician who served as the 39th President of the United States and was the recipient of the 2002 Nobel Peace Prize, the only U.S. President to have received the Prize after leaving office...
1979. - Awarded the Steele Prize by the American Mathematical Society. 1984.
See also
- Martingale (probability theory)Martingale (probability theory)In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
- Doob–Dynkin lemmaDoob–Dynkin lemmaIn mathematics, the Doob–Dynkin lemma, named after Joseph Doob and Eugene Dynkin, is a statement in probability theory that characterizes the situation when one random variable is a function of another, in terms of measurability and \sigma algebras....
- Doob martingaleDoob martingaleA Doob martingale is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration...
- Doob's martingale convergence theoremsDoob's martingale convergence theoremsIn mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph Leo Doob....
- Doob's martingale inequalityDoob's martingale inequalityIn mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time...
- Doob–Meyer decomposition theorem
External links
- A Conversation with Joe Doob
- Doob biography
- Record of the Celebration of the Life of Joseph Leo Doob
Articles by Doob
- Conditional brownian motion and the boundary limits of harmonic functions, Bulletin de la Société Mathématique de France, 85 (1957), p. 431–458.
- A non probabilistic proof of the relative Fatou theorem, Annales de l'institut Fourier, 9 (1959), p. 293–300.
- Boundary properties of functions with finite Dirichlet integrals, Annales de l'institut Fourier, 12 (1962), p. 573–621.
- Limites angulaires et limites fines, Annales de l'institut Fourier, 13 no. 2 (1963), p. 395–415.
- Some classical function theory theorems and their modern versions, Annales de l'institut Fourier, 15 no. 1 (1965), p. 113–135.
- Erratum : “Some classical function theory theorems and their modern versions” Annales de l'institut Fourier, 17 no. 1 (1967), p. 469–469.
- Boundary approach filters for analytic functions, Annales de l'institut Fourier, 23 no. 3 (1973), p. 187–213.
- Stochastic process measurability conditions, Annales de l'institut Fourier, 25 no. 3–4 (1975), p. 163–176.