Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate achieved by $L$ over the length of its path. For example, the path NENWSWW would have $\max(L) = 2$.
What is known about the distribution of $\max (L)$ over all paths of length $n$?
(It seems like this problem would have been studied, but most of the results on lattice paths I've found allow only north and east steps. Perhaps I'm not using the right search terms.)
Clarification: My question is that if we list all possible paths of length $n$, and sort them by the value of $\max (L)$ for each path, how many are there with $\max (L) = 0, \max (L) = 1,$ etc.? If we think about this from a random walk perspective, it's equivalent to imposing a uniform distribution on the four possible steps at each point.