# Central Limit Theorem The [Weak Law of Large Numbers](Laws%20of%20Large%20Numbers.md#Weak%20Law%20of%20Large%20Numbers%20WLLN) says that with high probability, $\frac{S_n}{n}$ is close to $\bar{X}$ for large $n$, but it establishes this via an upper bound on the tail probabilities rather than an estimate of what the distribution function, $F_{\frac{S_n}{n}}$ looks like. If we let our rv $\frac{S_n}{n}$ be normalized to have a mean of $0$ and unit variance, calling this normalized rv $Z_n$, we see that as $n$ increases this normalized rv seems to have its CDF converging to a specific distribution. This is indeed true! It is **converging in distribution** to the normalized Gaussian CDF.   --- Date: 20220803 Links to: Tags: #review References: * []()