The Central Limit Theorem for Independent Identically Distributed Square-Integrable Random Variables

In the setting of Gaussian Random Vectors: Standing Notation and Convergence of Probability Measures and Random Elements: Standing Notation, let $(X_{k})_{k\in\mathbb{N}}$ be an independent and identically distributed sequence of random variables in $\mathcal{L}^{2}(P)$ with $\operatorname{Var}X_{1}>0$, let $\mu=\mathbb{E}X_{1}$ and $\sigma=\sqrt{\operatorname{Var}X_{1}}$ (The k-th Root and the Square Root of a Nonnegative Real Number (ii)), and for $n\in\mathbb{N}$ let

$$T_{n}=\frac{1}{\sigma\sqrt{n}}\Bigl(\sum_{k=1}^{n}X_{k}-n\mu\Bigr).$$

(i) Central limit theorem. Let $Z$ be a standard normal random variable on a probability space $(\Omega',\mathcal{F}',P')$. Then

$$T_{n}\xrightarrow{d}Z.$$