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 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 processes, and
stationary processes, 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. 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. The postdoctoral members of the department of mathematics of the
University of Illinois are named J L Doob Research Assistant Professors. ==Honors==