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.
