Asymptotic Fluctuations of the Standard Random Walk

The standard random walk is easy to describe. Flip a fair coin with faces labelled ±1. If the face 1 shows up take a step of size 1 in the positive direction (along the x-axis) while if the face −1 shows up take a step of size 1 in the negative direction. We denote by Sn your location after n steps. The random walk is formally the sequence of random variables S0, S1, S2, . . . Throughout we assume that the walk starts at the origin of the x-axis, i.e. S0 = 0. If we think that we flip the coin once per unit of time, we can also interpret n as measuring time. This simple description makes the standard random walk amenable to combinatorial methods of investigation. The goal of this thesis is to describe a few less advertised asymptotic results concerning the fluctuations of the standard random walk. In the process we will reveal a few counterintuitive results that, surprisingly, occur in more general situations.