[M40] Let 12 ... be an infinite binary sequence, and let fn = 1 + 2 + + n. (a) A sequence is

[M40] Let ω1ω2 ... be an infinite binary sequence, and let fn =

ω1 + ω2 + ··· + ωn.

(a) A sequence ω is said to satisfy the infinite recurrence law if fn = 1 2n infinitely often. It can be shown that the set of infinite binary sequences having the infinite recurrence property has measure one in the set of all infinite binary sequences with respect to the usual binary measure. Show that there are infinite binary sequences that are Mises–Wald–Church random satisfying fn ≥ 1 2n for all n.

(b) Show that there are infinite binary sequences that are Mises–Wald–
Church random and satisfy lim supn→∞(
n i=1 ωi − 1 2n)/
√
n ln ln n >
1/
√
2 (they violate the law of the iterated logarithm), Exercise 1.10.5.
Comments. Since Items

(a) and

(b) are satisfied with probability zero in the set of all infinite binary strings, we can conclude that Mises–Wald–
Church random strings do not satisfy all laws of probability that hold with probability one (the laws of randomness). Source: [J. Ville, Etude ´
Critique de la Concept de Collectif, Gauthier-Villars, 1939].