5The Law of the Iterated Logarithm
The law of the iterated logarithm concerns the rate and oscillation of the almost sure convergence guaranteed by laws of large numbers.
In this chapter, we prove a game‐theoretic law of the iterated logarithm for a variant of Protocol 4.1, in which we proved Kolmogorov's law of large numbers. This game‐theoretic law says that
almost surely provided that the 
do not grow too fast and that 
, where 
.
Roughly speaking, (5.1) says that 
is almost surely the asymptotic least upper bound for the cumulative sum 
. By the symmetry between 
and 
in the protocol, it follows that is almost surely the asymptotic greatest lower bound for . We discuss only the upper ...
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