Let $\{X_{n}\}_{n\in \mathbb{N}}$ be a sequence of random variables. Suppose that $\{X_{n}\}_{n\in \mathbb{N}}$ fulfills certain regularity conditions described below. The random variables $X_{n}$ are independent and have finite means $\mathbb{E}(X_{n})$. In addition, $\sum_{n=1}^{\infty}{\rm var}(X_{n})<\infty$ and ${\rm var}(X_{n})=\sigma^2>0$ for at least one $n$.
QUESTION 1. What does it mean to say that the sum $\frac{\sum_{n=1}^{N}X_{n}}{N}$ deviates from its mean $\mathbb{E}\left( \frac{\sum_{n=1}^{N}X_{n}}{N}\right) $ macroscopically when $N\to\infty$? Does the term macroscopically have to do with some sequence of functions associated with the sequence of random variables $\{X_{n}\}_{n\in \mathbb{N}}$ being equicontinuous? Or does the term macroscopically have a statistical meaning or some physical meaning (coming from statistical mechanics)? Or is it just arbitrary terminology?
QUESTION 2. What does it mean to say that the sum $\frac{\sum_{n=1}^{N}X_{n}}{N}$ deviates from its mean $\mathbb{E}\left( \frac{\sum_{n=1}^{N}X_{n}}{N}\right)$ at the level of fluctuations when $N\to\infty$? Does it make any intuitive sense in terms of some sequence of graphs of the distribution functions $F_{S_{N}}$? Here $S_{N}= \frac{\sum_{n=1}^{N}X_{n}}{N}$.
This question came up after I watched the first lesson ( 03:50 to 04:15) on YouTube of the Probability II course taught by Professor Algustus Quadros Teixeira in 2020 at IMPA. I'm not a student at IMPA. I'm just watching the class yesterday. The lessons are not in English.
My attempt. It seems to me that the two expressions "deviating from the average macroscopically" and "deviating from the average in terms of fluctuations" have everything to do with the principle of large deviations. I looked for lecture notes and pdf materials on the internet that could explain these two expressions, such as this material from ArXiV. But I couldn't find them in any of the materials I searched for on the internet
Perhaps I wasn't able to correctly translate the expression from Portuguese into English. Please correct me if I have made any inaccuracies