3. Consider any Random Walk W = (Wn)n≥0 on Z, that is, Wn := a + X1 + · · · + Xn whereX1,X2,... areIIDRVs.
Let r := P0(return to 0) be the probability that W returns to 0 given it starts at 0. Let N be the total number of visits to 0 including the visit at time 0. Since intuitively the Random Walk ‘starts afresh’ whenever it first returns to 0, show that for r ∈ [0, 1),
P0(N =k)=rk−1(1−r) (k=1,2,...),
so that the total number of visits to 0 when started at 0 is a Geometrically dis- tributed RV. Also note that P0(N = +∞) = 1 when r = 1.
Hence, calculate the expected total number of visits to 0 when starting at 0, E0(N), in terms of r.
On the other hand, with ξn := I{Wn=0}, note N = ξ0 +ξ1 +ξ2 +... where the sum counts 1 for every time 0 is visited. By taking expectations and comparing expressions, deduce that
(see the picture)