Category: Publication

The result in this paper was included in L1-computability, layerwise computability and Solovay reducibility

News
Mar 12, 2012, Draft

Title
Weak L^1-computability and Limit L^1-computability

Type
Extended abstract

Journal
in preparation

Abstract
The class of the differences between two integral tests for Schnorr ran- domness is an important class related to Schnorr randomness. In this paper we study other randomness versions. We also claim that Solovay reducibility for lower semicomputable functions generalizes layerwise com- putability.

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in preparation

The result in this paper was included in L1-computability, layerwise computability and Solovay reducibility

News
Jan 26, 2012, Submitted to a conference

Title
An integral test for Schnorr randomness and its application

Type
Fullpaper

Journal
Submitted

Abstract
The author proposed in the previous paper that a characterization of a randomness notion by integral tests is a useful tool to study the relation between algorithmic randomness and computable analysis. In this paper we give a version of Schnorr randomness. With this result we show the connection between L1-computability and Schnorr layerwise computability. Finally we apply them to study the points on which two Radon-Nikodym derivatives are equal.

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preprint

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”.

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.