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Characterization of Kurtz randomness by a differentiation theorem

News
Dec 25, 2012, Published
Aug 8, 2012, Published online
July 12, 2012, Accepted with a minor revision
Dec 23, 2011, Need to fix it
Oct 18, 2011, Submitted to a Journal

Title
Characterization of Kurtz randomness by a differentiation theorem

Type
Fullpaper

Journal
Theory of Computing Systems: Volume 52, Issue 1 (2013), Page 113-132
The page in TOCS

Abstract
Brattka, Miller and Nies showed that some major algorithmic randomness notions are characterized via differentiability.
The main goal of this paper is to characterize Kurtz randomness by a differentiation theorem on a computable metric space.
The proof shows that characterization by integral tests plays an essential part and shows that how randomness and differentiation are connected.

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preprint

Posted on

The law of the iterated logarithm in game-theoretic probability

News
15 Nov 2012, the revised slides were uploaded.
26 Oct 2012, the slides were uploaded.

Title
The law of the iterated logarithm in game-theoretic probability

Type
Fourth Workshop on Game-Theoretic Probability and Related Topics
12-14 Nov 2012
The University of Tokyo

Abstract
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.

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Slides