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Almost Sure Convergence

Definition (Almost Sure Convergence)

Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. Let (Xn)n∈N(X_{n})_{n\in\mathbb{N}} be a sequence of random variables, and XX be another RV. We say XnX_n converges with probability one (w.p.1) or Almost surely to X if ∃A∈F:P(Ac)=0:∀ω∈A:Xn(ω)→X(ω)\exists A\in\mathcal{F}:P(A^{c})=0:\forall\omega\in A:X_{n}(\omega)\to X(\omega) or P({ω∈Ω: lim⁡n→∞Xn(ω)=X(ω})=1P(\{\omega\in\Omega: \ \lim_{n\to\infty}X_n(\omega)=X(\omega\})=1 Write Xn→a.s.X.X_n\xrightarrow{a.s.} X.

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Radner Krainak Theorem

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Metropolis Hastings Algorithm

Stationary & Ergodic Source

Law of Large Numbers

Scheffé's Theorem

Skohorod's Theorem

Theorems on Convergence

Distribution

Summary of MATH 895

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Itô's Formula

Multidimensional Itô Formula

Quadratic Variation

Harris Recurrent

Positive Recurrent

Transient

Doob's Forward Convergence Theorem

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Lévy's Convergence Theorems

Martingale Convergence Theorem