May 4, 2018. Accepted by AOP

**Title**

Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach

(with T. Sasai and A. Takemura)

**Type**

Full paper

**Journal**

Annals of Probability,

to appear.

**Abstract**

We prove an Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. As many other game-theoretic proofs, our proof is self-contained and explicit.

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arXiv

8 Aug, 2017. Online

28 July 2017. accepted by SPA

**Title**

Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

(with R. Sato and A. Takemura)

**Type**

Full paper

**Journal**

Stochastic Processes and their Applications

Volume 128, Issue 5, May 2018, Pages 1466-1484

The page at SPA

**Abstract**

We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion

betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We

establish the relation between the rate of convergence of the strong law of large numbers in the selfnormalized

form and the rate of divergence to infinity of the prior density around the origin. In

particular we present prior densities ensuring the validity of Erdos–Feller–Kolmogorov–Petrowsky ˝

law of the iterated logarithm.

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arXiv

19 Mar 2018, online

5 Mar 2018, accepted by JLA

**Title**

Randomness and Solovay degrees

(with A. Nies and F. Stephan)

**Type**

Full paper

**Journal**

Journal of Logic and Analysis, Vol 10, pp.1–13, 2018.

Open access

**Abstract**

We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-L\”of’s, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-L\”of-randomenss, we show that Schnorr randomness is not upward closed in the Solovay degrees. Next, some left-r.e. Schnorr random $\alpha$ is the sum of two left-r.e. reals that are far from random. We also show that the left-r.e. reals of effective dimension $>r$, for some rational $r$, form a filter in the Solovay degrees.

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coromandel_JLA_final

11 Mar 2018, Submitted

**Title**

Computable Measure Theory

**Type**

Survey in Japanese

**Journal**

RIMS Kokyuroku 25 – 27 Dec 2017

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rims

11 Mar 2018, submitted to a Journal

**Title**

Muchnik degrees and Medvedev degrees of the randomness notions

**Type**

Full paper

**Journal**

TBA

**Abstract**

The main theme of this paper is computational power when a machine is allowed to access random sets.

The computability depends on the randomness notions and we compare them by Muchnik and Medvedev degrees.

The central question is whether, given an random oracle, one can compute a more random set.

The main result is that, for each Turing functional,

there exists a Schnorr random set whose output is not computably random.

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Muchnikdegrees

1 Mar, 2018 The slide file was uploaded.

**Title**

Continuity of limit computable functions

**Type**

A talk in undergraduate colloquium in University of Hawaii at Manoa

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hawaii

24 Feb, 2018 The slide file was uploaded.

**Title**

Continuity of limit computable functions at 1-generic points

**Type**

A talk in a seminar with Suzuki lab(in Japanese)

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tmu

24 Dect, 2017 The slide file was uploaded.

**Title**

A hierarchy of functions corresponding randomness hierarchy

**Type**

A talk in RIMS workshop “Proof theory and proving”

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rims

13 Dect, 2017 The slide file was uploaded.

**Title**

Computation at random points

**Type**

A talk in IPA Math 2017

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IPAmath

13 Dec, 2017 The slide file was uploaded.

**Title**

What is mathematics at all?

**Type**

A lecture in Meiji University Meiji High School at 6 Dec, 2017.

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math