Title
Characterization of Kurtz randomness by a differentiation theorem
Type
Talk
This talk is in CS Seminar at Kyoto University on 17 Nov 2011.
The same talk will be given in ALC 2011.
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宮部賢志(ミヤベケンシ)
Title
Characterization of Kurtz randomness by a differentiation theorem
Type
Talk
This talk is in CS Seminar at Kyoto University on 17 Nov 2011.
The same talk will be given in ALC 2011.
Download
News
Oct 25, 2011, Available on line
Oct 18, 2011, Accepted
April 5, 2011, Submitted
Title
Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability
Type
Fullpaper
Conference and Journal
Stochastic Processes and their Applications, 122:1-30, 2012.
Journal
arXiv
Abstract
We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001). We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality’s strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results, which can be interpreted in measure-theoretic framework.
You can read a comment by Vovk at Working Papers in
the website of “Probability and Finance”.
Title
Randomness and differentiability
Type
Talk
This talk is in WORKSHOP ON PROOF THEORY AND THEORY OF COMPUTING 2011 at Tokyo Metropolitan University on Sep 12, 2011.
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slide
Title
An optimal semimeasure and its convergence
Type
Talk
This talk is in ALGI 22 at University of Tokyo on Aug 23, 2011.
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slide
News
July 16, 2011, Accepted.
June 16, 2011, Resubmitted to a Journal
Mar 29, 2011, Submitted to a Journal
Title
The diifference between optimality and universality
(Former)Degree of non-randomness and uniform Solovay reducibility
Type
Fullpaper
Journal
Logic Journal of the IGPL (2012) 20 (1): 222-234.
abstract page
Abstract
We introduce a degree of non-randomness using a test concept.
The degree implies a notion of an optimal Martin-L\”of test, which is different from a universal test.
In the latter half we generalize Solovay reducibility.
Solovay reducibility is a measure of relative randomness between two reals.
We introduce uniform solovay reducibility, which is a measure of relative randomness between two sequences of reals.
Finally we prove that a sequence is uniform Solovay complete iff it is the sequence of measures of an optimal Martin-L\”of test.