This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details.(November 2021) (Learn how and when to remove this message)
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob,[1] also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time.
When analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are martingales and Azuma's inequality.[clarification needed]
^
Doob, J. L. (1940). "Regularity properties of certain families of chance variables" (PDF). Transactions of the American Mathematical Society. 47 (3): 455–486. doi:10.2307/1989964. JSTOR 1989964.
mathematical theory of probability, a Doobmartingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates...
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...
sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at...
In probability theory, a martingale difference sequence (MDS) is related to the concept of the martingale. A stochastic series X is an MDS if its expectation...
Wiener process. Corollary. (See also Doob'smartingale convergence theorems) Let Mt be a continuous martingale, and M ∞ − = lim inf t → ∞ M t , {\displaystyle...
Z ) {\displaystyle E(E(X\mid {\mathcal {H}})\mid Z)=E(X\mid Z)} . Doobmartingale property: the above with Z = E ( X ∣ H ) {\displaystyle Z=E(X\mid {\mathcal...
particularly martingales, and the mathematical field of potential theory, with early ideas by Shizuo Kakutani and then later work by Joseph Doob. Further...
M\rangle } is a local martingale. Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as...
to martingale-Hp. Doob's maximal inequality implies that martingale-Hp coincides with Lp(Ω, Σ, P) when 1 < p < ∞. The interesting space is martingale-H1...
way of bounding the differences by applying Azuma's inequality to a Doobmartingale. A version of the bounded differences inequality holds in the matrix...
(Lebesgue) almost everywhere convergence of Fourier series of L2 functions Doob'smartingale convergence theorems a random variable analogue of the monotone convergence...
Global Information