2.4 Strong law of large numbers (Borel)

Let $ \left\{ \mathbf{X}_{n}\right\} $ be a sequence of identically distributed random variables with mean $ \mu $ and variance $ \sigma^{2} $ , and $ Cov\left(\mathbf{X}_{i},\mathbf{X}_{j}\right)=E\left[\left(\mathbf{X}_{i}-\mu\right)\left(\mathbf{X}_{j}-\mu\right)\right]=0,\quad i\neq j\text{ : uncorrelated.} $

Then $ \mathbf{Y}_{n}=\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k}\longrightarrow\left(a.e.\right)\longrightarrow\mu\text{ as }n\longrightarrow\infty. $

Proof

Beyound this course. Require measure theory.


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