Given a 1D random walk (simple +1, -1 movements from the axis) I've seen proofs that the expected absolute distance tends to Sqrt(2*n/PI) and I've plotted graphs of 1D random walks along with this curve along and the associated standard deviation of the absolute distance. As expected most walks can be found in the area around the expected absolute distance curve up to 2 standard deviations from it.
Now take a walk that zooms steeply up or down towards the 2 standard deviation line from the expected absolute distance curve. From what little I understand of statistics very few random walks should be found above this line - 95% rule (and indeed my graphs confirm this). So doesn't this imply that such a walk should therefore most likely turn around and head back into the area closer to the expected absolute distance curve? Or at least continue on its original direction but at a less steep path thus following the root(N) curve closely. But this seems silly because it implies to some degree that you can predict the motion of the path (either a bounce or a flattening of the trajectory). Hence my confusion!