Kolmogorov Strong Law of Large Numbers

Last updated Nov 1, 2022

# Statement

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space

and let $(X_{n}){n=1}^{\infty}$ be a Sequence

Sequence

Definition A $f: \mathbb{N} \to X$ for some $X$ . It is usually denoted $\{xn\}{n=1}^{\infty} \subset X$ or $(x{n}) \subset...

11/7/2022

of iid Random Variable

Random Variable

Definition Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a . A is a $X: \Omega \to \mathbb{R}$ . Remarks Rather than say $X$ is...

11/7/2022

s on $\Omega$ . Let $S{n} = \sum\limits_{i=1}^{n} X_{n}

. Then there exists

c \in \mathbb{R}

s.t.

\frac{S_{n}}{n} \to c$ a.s. If and Only If

\mathbb{E}|X_{1}| < \infty

, in which case

c = \mathbb{E}(X_{1})

# Proof

TODO

- See thm 7.5.1 in Resnick - A Probability Path pg 220

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Kolmogorov Strong Law of Large Numbers

# Statement

Probability Space

Sequence

Random Variable

Almost Sure Convergence

If and Only If

# Proof

TODO

Resnick - A Probability Path

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