Consider an infinite series of independent flips of a fair coin. On the nth flip, count up the total number of heads and call that Sn.There are classic theorems in probability that can tell us about the behavior of Sn. In particular the Central Limit Theorem tells us about the long-term distribution. The Strong and Weak Laws of Large Numbers, tell us about convergence to the mean. The Law of the Iterated Logarithm, established by Khinchin and Kolmogorov in the 1920s, tells us about the degree and

frequency of extreme deviations from the mean as our series of coin flips progresses. In this talk, we will see that |𝑆_n− 𝑛/2| has an almost sure upper bound at \lambda\sqrt{n/2 loglog(n)} for any \lambda>1, which it may exceed only finitely often. Any oscillation less than \sqrt{n/2 loglog(n)} will occur infinitely often. 

The intuition for this upper bound, and the obscure loglog factor can be found in the proof, which I will outline.