Central Limit Theorem

The Central Limit Theorem (CLT) states that given a sum distribution \(S_n\) as a sum of \(n\) iid random variables \(X\) of mean \(\mu\) and standard deviation \(\sigma\),

\[\begin{split} \begin{gather*} S_n = X_1 + X_2 + \ldots + X_n\\ E(X_n) = n\mu\\ \text{SD}(X_n) = \sqrt{n}\sigma \end{gather*} \end{split}\]

As \(n \to \infty\), \(S_n\) approaches normal distribution.

\[ S_n \to \text{Normal}(N\mu, N\sigma^2) \]

Concentration Corollary : Since the distribution of the sum tends to normal at large \(N\), its CDF as well,

$$
\begin{gather*}
P(S_n \le s) \approx \Phi\left(\frac{s-N\mu}{\sqrt{N}\sigma}\right)
\end{gather*}
$$

More used is the central bulk concentration,

$$
\begin{align*}
P(\abs{S_n - N\mu} \le c) &= P(S_n \le N\mu + c) - P(S_n \le N\mu - c)\\
&\approx \Phi\left(\frac{c}{\sqrt{N}\sigma}\right) - \Phi\left(\frac{c-2N\mu}{\sqrt{N}\sigma}\right)
\end{align*}
$$