[M42] Consider a sequence generated by a (p, 1 − p) Bernoulli process. Show that lim supn→∞ S∗

n/

√

2 ln ln n = 1 with probability one.

Comments. For reasons of symmetry, lim infn→∞ S∗

n/

√

2 ln ln n = −1.

This remarkable statement, known as the law of the iterated logarithm is due to A.I. Khintchin [Fundamenta Mathematicae 6(1924), 9–20] and was generalized by A.N. Kolmogorov [Math. Ann., 101(1929), 126–135].

For an explanation of its profundity, implications, and applications see also [W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968].