For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$.

What is the bound, in terms of $n$, such that if the walk goes for $n$ steps, with constant probability it *always stays* within that bound? It is probably no longer on the order of $\sqrt{n}$, but is it still less than the order of $n$?