An Elementary Proof of the Hitting Time Theorem

In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a left-continuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at time n, is equal to k/n. Here, a walk is called left-continuous when its steps are bounded from below by −1. We start by introducing some notation. Let Pk denote the law of a random walk starting at k ≥ 0, let {Yi}∞i=1 be the independent and identically distributed (i.i.d.) steps of the random walk, let Sn = k + Y1 + · · · + Yn be the position of the random walk starting at k after n steps, and let T0 = inf{n: Sn = 0} (1) denote the walk’s first hitting time of the origin. Clearly, T0 = 0 when the walker starts at the origin. Then, the hitting time theorem is the following result: Theorem 1 (Hitting time theorem). For a random walk starting at k ≥ 1 with i.i.d. steps {Yi}∞i=1 satisfying Yi ≥ −1 almost surely, the distribution of T0 under Pk is given by