[marv]

Assume X_i = +1 or -1. If we can look into the future, just pick N = least s.t. X_{N+1} != X_N.

Then Z_N = sqrt(N) X_1. As N and X_1 are independent, E(Z_N) = 0 and Var(Z_N) = E((Z_N)^2) = E(N) = sum{k>=1} k/2^k = 2.

Marv