In this video CLICK HERE, the professor says at 51:19 that if we add n Gaussian random variables and we see that sum further and further from our eyes, the result will look like a Brownian motion.
I do not not really understand what he means. I know that adding Gaussian random variables will be Gaussian and even if they are not originally Gaussian, the result will be Gaussian if n is high enough (by the central limit theorem). So I can not see any explanation on the internet that says that if we look further at the sum of Gaussian random variables, the result would look like a Brownian motion. Is there any interesting link that you can share with me concerning this fact?
Any help will be very appreciated!
The word "add" (or "sum") is being used imprecisely here. You can construct an approximation to a Brownian motion by starting with $n$ independent standard Gaussian variables $X_1,\ldots,X_n$. Define the partial sums $$ S_0:=0,\quad S_1:=X_1, \quad S_2:=X_1+X_2,\ \ldots,\ S_k:=\sum_{i=1}^kX_i,\ \ldots,\ S_n:=\sum_{i=1}^nX_i. $$ Then plot $S_k$ versus $k$ in the plane, and connect the dots. This creates a continuous function. Under suitable scaling of the $x$ and $y$ axes, this function converges to a Brownian motion as $n\to\infty$. "Looking at this from farther and farther" is an informal way of saying "under suitable scaling".
There are many ways to construct Brownian motion: see Reference for the Construction of Brownian Motion. Donsker's Theorem, mentioned in the video, justifies the construction of Brownian Motion as the scaled limit of a random walk.