The maximum of a symmetric next neighbor walk on the nonnegative integers

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution ¥[Sn = x, maxi<7< Sn = a] of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for Sn - x, but more importantly that for maxi<7< Sj = a asymptotically at fixed a2/n . We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.