mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales...

condition (c) the stopped process Xτ is bounded, hence by Doob's martingale convergence theorem it converges a.s. pointwise to a random variable which we call...

577. Martingale (probability theory) Doob–Dynkin lemma Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Doob–Meyer...

and Convergence. Wiley. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Doob–Meyer...

theorem establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions Doob's martingale convergence theorems a...

theorem (asymptotic analysis) Donsker's theorem (probability theory) Doob decomposition theorem (stochastic processes) Doob's martingale convergence theorems...

Wiener process. Corollary. (See also Doob's martingale convergence theorems) Let Mt be a continuous martingale, and M ∞ − = lim inf t → ∞ M t , {\displaystyle...

rule Donsker's theorem Doob decomposition theorem Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Doob–Meyer decomposition...

uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions...

For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007). Theorem—Let a martingale M n {\textstyle M_{n}} satisfy 1 n ∑...

_{k}^{i}\,dB_{s}^{k},} whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of X t {\displaystyle...

Limit theorems Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation...

{\displaystyle X_{t}} is also a Gaussian process. In other cases, the central limit theorem indicates that X t {\displaystyle X_{t}} will be approximately normally...

càdlàg martingales M such that E[(M*t)p] is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities...

Limit theorems Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation...

Continuous stochastic process / (U:RG) Doob's martingale convergence theorems / (SU:R) Doob–Meyer decomposition theorem / (U:R) Feller-continuous process /...

extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process. Suppose...

so they are often used to derive convergence results, due largely to martingale convergence theorems. Martingales have many applications in statistics...

cards. Snell earned his Ph.D. in 1951 ("Applications of Martingale System Theorems"), with Doob as his supervisor. At Dartmouth College Snell became involved...

the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Suppose...

Limit theorems Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation...

Piecewise-deterministic Markov process Martingale Doob martingale Optional stopping theorem Martingale representation theorem Azuma's inequality Wald's equation...

some function f by the martingale convergence theorem. Moreover, Mn converges to f in Lp-norm by the dominated convergence theorem; hence Mn can be expressed...

Mises Theorem". Asymptotic Statistics. Cambridge University Press. ISBN 0-521-78450-6. Doob, Joseph L. (1949). "Application of the theory of martingales"....

the study of vector-valued martingales, with the first proof of the ‘strong’ almost everywhere convergence for martingales taking values in a Banach space...

best known for his continuous-time analog of Doob's decomposition of a submartingale, known as the Doob–Meyer decomposition and his work on the 'general...

{H}}))+\operatorname {Var} (\operatorname {E} (X\mid {\mathcal {H}}))} . Martingale convergence: For a random variable X {\displaystyle X} , that has finite expectation...

wide-ranging research included the analysis of martingale inequalities, Markov processes, de Finetti's theorem, consistency of Bayes estimators, sampling...

E[S_{t+1}|S_{t}]=S_{t}} which follows a martingale, the EMH does not always assume that stocks follow a martingale. Research by Alfred Cowles in the 1930s...