Today we will continue to discuss random walks. In the previous post, we were proving Skorokhod embedding theorem. We introduced a random variable for a simple random walk that takes a value x greater than the value taken by random variable A but not reaching the value taken by the random variable B and assuming that this walk is independent of both A and B. The claim was that this new random variable takes a probability pk from the distribution and thus leading to the said theorem that the stopping time takes a probability from the distribution. To make the claim, we looked at the initial values when k = 0, then Z = 0 if and only if A = 0 and B = 0 and this has probability p0 by definition. Then we take k > 0 and in this case we work out the probability that our random variable takes the value k.  We first expand this probability as the cumulative sum of the independent probabilities over all i < 0 and j > 0 . The independent probabilities are for the random variable to take a value k and for A to take value i and B to take value j.  We can eliminate j let B take value k for the probability we are calculating. Then we apply the theorem from the gambler's ruin that describes the probability to take value b > x before reaching a  < x as equals (x - a) / (b - a). Further we use the modified expression of that probability in terms of a random walk Ta ^ Tb  to reach level b as the same probability and equaling  (-a / (b-a)) or (-i/(k-i)) as in this case.  So we simplify the independent probabilities with this value and the normalization factor times (k-i) pi pk. Since the sum of this has a zero average component, it simplifies to the probability pk thus proving the theorem.