Category: Publication

Posted on

Uniform Kurtz randomness

News
4 Nov 2013, Published online
16 May 2013, Submitted to a Journal

Title
Uniform Kurtz randomness
(with Takayuki Kihara)

Type
Fullpaper

Journal
Journal of Logic and Computation, 24 (4): 863-882, 2014
doi: 10.1093/logcom/ext054

Abstract
We propose studying uniform Kurtz randomness, which is the uni- form relativization of Kurtz randomness. This notion has more natural properties than the usual relativization. For instance, van Lambalgen’s theorem holds for uniform Kurtz randomness while not for (the usual relativization of) Kurtz randomness. Another advantage is that lowness for uniform Kurtz randomness has many characterizations, such as those via complexity, martingales, Kurtz tt-traceability, and Kurtz dimensional measure.

Download
preprint

Posted on

An optimal superfarthingale and its convergence over a computable topological space

News
Oct, 2013. Online
Sep 3, 2011. Accepted
June 7, 2011. Submitted

Title
An optimal superfarthingale and its convergence over a computable topological space

Type
Conference paper

Conference
Solomonoff 85th Memorial Conference at Melbourne, Australia.

Lecture Notes in Computer Science
Volume 7070 2013
Algorithmic Probability and Friends. Bayesian Prediction and Artificial Intelligence
Papers from the Ray Solomonoff 85th Memorial Conference, Melbourne, VIC, Australia, November 30 – December 2, 2011
http://link.springer.com/book/10.1007/978-3-642-44958-1

Download
preprint

Abstract
We generalize the convergenece of an optimal semimeasure
to a real probability in algorithmic probability by using game-theoretic
probability theory and the theory of computable topology. Two lemmas
in the proof give as corollary the existence of an optimal test and an
optimal integral test, which are important from the point of view of
algorithmic randomness. We only consider an SCT3 space, where we
can approximate the measure of an open set. Our proof of almost-sure
convergence to the real probability by a superfarthingale indicates why
the convergence in Martin-L¨of sense does not hold.

Posted on

The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges

News
19 Mar 2013, Accepted
25 Aug 2012, Submitted to a Journal

Title
The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges
(with Akimichi Takemura)

Type
Fullpaper

Journal
Stochastic Processes and their Applications, 123, 3132-3152, 2013.

Abstract
We prove both the validity and the sharpness of the law of the iter- ated logarithm in game-theoretic probability with quadratic and stronger hedges.

Download
arXiv
preprint