Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 3 / b / iv

Intersecting the probability-one hitting events over all positive integers shows that the simple symmetric random walk reaches arbitrarily large positive levels almost surely. The hitting times of levels tending to infinity also tend to infinity, since $|S_n|\le n$. Hence $\mathbb{P}(B_{+})=1$, and in particular $\mathbb{P}(B_{-}\cup B_{+})=1$.

Alternatively, the union conclusion, the equal probabilities from symmetry, and the Kolmogorov zero-one law force both probabilities to equal one. Thus almost every path has arbitrarily large positive and negative excursions. A nearest-neighbor integer path must pass through zero between a positive and a negative excursion. Such crossings occur after every finite time, givingThis is recurrence of the simple symmetric random walk.