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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 ...
Jun 7, 2004 · In 1942 Veblen approached him to work in Washington for the navy on mine warfare. He worked there until 1945 when he returned to the University of Illinois. Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory. The paper [1] looks at many of the areas of probability ...
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Doob decomposition theorem. In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero.
Doob’s was a brilliant but daunting compilation of technical facts, unembellished by examples. Doob’s goal was to show that probability theory could stand with any other branch of mathematical analysis, and he succeeded. The fact that most young probabilists have never read Doob’s book can be seen as a testament to its success.
Doob’s essential contributionsto Probability theoryare discussed; this includes the main early results on martingale theory, Doob’s h-transform, as well as a summary of Doob’s three books. Finally, Doob’s ‘stochastic triangle’ is viewed in the light of the stochastic analysis of the eighties. 1. BiographyofJ.L.Doob:Somekeypoints.
Doob’s approach to probability was evident in his first probability paper (see Doob (1934)). Doob proved theorems related to the law of large numbers, using the fact that the law of large numbers follows from a probabilistic interpretation of Birkhoff’s ergodic theorem.
Doob’s Optional Stopping Theorem: statement. I. Doob’s Optional Stopping Theorem: If the sequence X. 0;X. 1;X. 2;::: is a bounded martingale, and T is a stopping time, then the expected value of X. T. is X. 0. I. When we say martingale is bounded, we mean that for some C, we have that with probability one jX. i. j< C for all i. I. Why is ...
