15 Nov 2012, the revised slides were uploaded.
26 Oct 2012, the slides were uploaded.
The law of the iterated logarithm in game-theoretic probability
Fourth Workshop on Game-Theoretic Probability and Related Topics
12-14 Nov 2012
The University of Tokyo
The Kolmogorov law of the iterated logarithm (LIL) (1929) provides the exact speed of the convergence of the sum of independent ran- dom variables under a condition. Subsequently, Hartman and Wintner (1941) showed that, in the case of i.i.d. random variables, the existence of a second moment is sufficient for the LIL. Shafer and Vovk (2001)
studied the Kolmogorov LIL in game-theoretic probability and asked the treatment of the Hartman and Winter LIL in game-theoretic prob- ability.
I present a new sufficient condition for the LIL in game-theoretic probability, which has a similar form to the Hartman and Winter LIL. The main idea is to add a little stronger hedges. This is the joint work with Akimich Takemura.