What version of the Central Limit Theorem is this?

Question

In lectures the following was written down on the board

$$\mathbb{P}\left(\frac{X_1+\dots + X_n}{\sqrt{n}}>X\right)$$

Obviously this is in an incomplete state, and I all I managed to catch was that this had something to do with the Central Limit Theorem.

I've been unable to match this with any of the versions of the theorem and would appreciate even a link to the version which this incomplete state refers to.

For the sake of context, this appeared during a discussion of the following scenario:

Suppose you place the tip of your pen on a piece of paper. The ink will spread out from it onto the paper and over a long period of time, almost surely converge to a circle

I assume that this is because of the Central Limit Theorem?

Answer 1

Theorem: If $X_1,X_2,X_3,\ldots$ are i.i.d. with expected value $0$ and variance $1$ then $$ \lim_{n\to\infty} \Pr\left( \frac{X_1+\cdots+X_n}{\sqrt n} > x \right) = 1 - \Phi(x) $$ where $\Phi$ is the cumulative probability distribution function of the standard normal distribution, given by $$ \Phi(x) = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^x e^{-u^2/2}\,du. $$

Answer 2

Assuming the $X_i$ are i.i.d. with $\Bbb{E}(X_i) = 0$ and finite variance, $\sigma^2$, $$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i \right) \xrightarrow{dist} \mathcal{N}(0,\sigma^2) \text{.}$$ (This is the Lindeberg-Levy CLT.) The thing on the left is your $\dfrac{X_1 + \cdots + X_n}{\sqrt{n}}$, so your expression is just talking about right tails of a normal distribution.