Weak Law of Large Numbers

In the setting of Probability Spaces: Notation, let $\mu$ and $c\ge0$ be real numbers, and let $(X_{k})_{k\in\mathbb{N}}$ be a sequence of random variables in $\mathcal{L}^{2}(P)$ such that $X_{j}$ and $X_{k}$ are uncorrelated for all $j,k\in\mathbb{N}$ with $j\neq k$, and

$$\mathbb{E}X_{k}=\mu,\qquad\operatorname{Var}X_{k}\le c\qquad(k\in\mathbb{N}).$$

(i) Chebyshev bound. For every $n\in\mathbb{N}$ and every real $\varepsilon>0$,

$$P\Bigl(\Bigl|\frac{1}{n}\sum_{k=1}^{n}X_{k}-\mu\Bigr|\ge\varepsilon\Bigr)\le\frac{c}{n\varepsilon^{2}}.$$

(ii) Weak law. For every real $\varepsilon>0$,

$$P\Bigl(\Bigl|\frac{1}{n}\sum_{k=1}^{n}X_{k}-\mu\Bigr|\ge\varepsilon\Bigr)\to0\qquad(n\to\infty).$$