Note:

Can you guess what happens in general when we have , and the movements are  units to right,  units to left,  units to up and  units to down, such that, . Comment intelligently on this problem.

Again let us continue with the problem which we were discussing. So we have

as

Using Stirling's formula or approximation, which is , we have . Again let us pay attention to the fact that  when , hence the rate of convergence for  is not zero, else the rate of convergence of  is 0. So now we have the sequence , , …., and the sum, i.e.,  iff . Thus the two dimension random walk is recurrent iff .