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Almost Sure Convergence

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Definition
ProbabilityStochasticProcesses

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.

Linked from

Beppo Levi Theorem

Controlled Markov Chain

Blackwell's Irrelevant Information Theorem

Belief MDP

Filter Stability

Kalman Filter

Q-Learning

Optimality of Q Learning Algorithm

Four Hypotheses

Static Quadratic Team

Existence for N-agent Team

Uniqueness for N-agent Team

Radner Krainak Theorem

Stationary Radner Krainak Theorem

Stationary & Ergodic Source

Law of Large Numbers

Scheffé's Theorem

Skohorod's Theorem

Theorems on Convergence

Distribution

Summary of MATH 895

Exponential Random Variable

Itô's Formula

Multidimensional Itô Formula

Quadratic Variation is CAII

Metropolis Hastings Algorithm

Harris Recurrent

Positive Recurrent

Transient

Lp Martingale

Doob's Forward Convergence Theorem

Lévy's Convergence Theorems

Martingale Convergence Theorem