I will be learning about stochastic processes during my next semester and I wanted to get a headstart by reading up on simple random walks.
From the "One-dimensional random walk" section of the Wikipedia page here, a simple random walk on $\mathbb{Z}$ is defined to be the series $\{S_n\}$, where $S_0 = 0$, $S_n = \sum\limits^n_{i = 1} Z_i$ and $Z_i$ are independent random variables taking on the values $1$ or $-1$, each with probability $\frac 1 2$. It then goes on to find that $\mathbb{E}[S_n] = 0$ and $\mathbb{E}[S_n^2] = n$.
However, what I am unclear is the next part, which says "This hints that $\mathbb{E}[\lvert S_n \rvert]$, the expected translation distance after $n$ steps, should be of the order $\sqrt n$. In fact, $\lim\limits_{n \rightarrow \infty} \frac {\mathbb{E}[\lvert S_n \rvert]} {\sqrt n} = \sqrt {\frac 2 \pi}$.".
After pondering about this, the following two questions came to mind:
How does the previous part hint to the fact that $\mathbb{E}[\lvert S_n \rvert] \approx \sqrt n$? I know that, since $\mathbb{E}[S_n^2] = n$, we have $\sqrt {\mathbb{E}[S_n^2]} = \sqrt n$, but $\sqrt {\mathbb{E}[S_n^2]} \neq \mathbb{E}[\lvert S_n \rvert]$ right?
How did they end up with that limit and what is the significance of that limit?
As mentioned above, this is just some independent learning I have decided to take on during my summer break and I have not formally covered random walks yet, so any intuitive explanations will be greatly appreciated :) I also apologise in advance if these are trivial questions, or if they have been asked before!