L^1-computability and weak L^1-computability

Title
L^1-computability and weak L^1-computability

Type
Talk at “Kyoto computable analysis Symposium 2012” on 24-27 Feb 2012.

Abstract
Computable functions are simple functions and
in Weihrauch approach computable functions are always continuous.
However there are some simple discontinuous functions such as the
floor function.
Therefore we need another mathematical notion to measure simplicity.
One candidate is L^1-computability, which was introduced by Pour-El
and Richard 1989.
In this talk I show you that more effectivised version of L^1-computability
has a strong connection with Schnorr randomness
and that some weaker versions of L^1-computability has connections
with some stronger randomness notions.

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Schnorr Layerwise Computability

Title
Schnorr Layerwise Computability

Type
Talk at “Workshop on Proof Theory and Computability Theory 2012” on 20-23 Feb 2012.
Workshop website

Abstract
In order to formalize the notion of randomness mathematically,
the theory of algorithmic randomness uses computability theory.
Recent researches show that some notions in algorithmic randomness conversely
are useful to study computable analysis.
One example is layerwise computability defined by Hoyrup and Rojas 2009.
In this talk I introduce Schnorr layerwise computability,
which is a Schnorr-randomness version,
and explain why this is a more natural notion.

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An integral test for Schnorr randomness and its application

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

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An introduction to algorithmic randomness

Title
An introduction to algorithmic randomness

Type
Talk at GCOE tea time on 10 Jan 2012.

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slides

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Characterization of Kurtz randomness by a differentiation theorem

Title
Characterization of Kurtz randomness by a differentiation theorem

Type
Talk, invited as a special session speaker

Twelfth Asian Logic Conference
15-20 December 2011, Wellington

The talk is based on the paper which has the same title.

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Randomness and separation axioms

Title
Randomness and separation axioms

Type
Talk

ANALYSIS AND RANDOMNESS IN AUCKLAND
DEC 12 AND 13, 2011

The talk is based on the paper of “Algorithmic randomness over general spaces”.

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An optimal superfarthingale and its convergence over a computable topological space

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

Type
Talk
Solomonoff 85th Memorial Conference, (30 Nov – 2 Dec 2011), Melbourne

The talk is based on the paper which has the same title.

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slides

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Characterization of Kurtz randomness by a differentiation theorem

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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Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

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”.

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Randomness and differentiability

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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