So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative sums of random variables. Formally:
We are given some finite parameters $M,N \in \mathbb{N}$
Let $X_i \sim U(-M,M)$ for $i \in \{1,2,\dots,N\}$. In both cases of a continuous or discrete uniform I would be quite happy to find a solution, so you can assume whatever you prefer.
Let $Y_i=\sum_{j=1}^i X_j$ for $i \in \{1,2,\dots,N\}$
$E[\min(Y_i)]=?$
The only solution so far I found is simulation, but I was wondering for any approach for a analytical solution.