Here is the question:
For $n = 25$ and $50$, approximate the probability $P(max_{1 \leq k \leq n} S_k > 2\sqrt n)$ when the sample observations are iid $U[−1, 1]$, where $S_k= X_1 + X_2+...+X_k$.
I understand that it wants us to approximate the probability that the maximum cumulative sum of the $X_i$s is greater than $2\sqrt n$. So, I created a code in R to run the simulation 10000 times and the probability turns out to be very small (i.e 5e-04, 3e-04).
However, I am stuck on how this can be shown by hand. Any help is greatly appreciated!
I also found this theorem in my book that might be related but I am not sure because it has n going to infinity:
$$\lim_{n\rightarrow \infty} P(max_{1 \leq k \leq n} S_k \leq x\sqrt n) = G(x), \text{ where } G(x)= 2\Phi(x)-1. $$