8. Consider the two-dimensional random walk of Exercise 10.11, in which a particle inhabits the integer points {(i, j ) : i, j = . . . ,−1, 0, 1, . . . } of the plane, moving rightwards, upwards, leftwards or downwards with respective probabilities p, q, r , and s at each step, where 10.5 Problems 179 p, q, r, s > 0 and p+q+r +s = 1. Let Sn be the particle’s position after n steps, and suppose that S0 = (0, 0). Let vn be the probability that Sn = (0, 0), and prove that v2m =



2m m

2 

1 4

2m if p = q = r = s = 1