8 Dec, 2018 The slide file was uploaded.

**Title**

Construction of random and nonrandom sets

**Type**

A talk in Sendai Logic School

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SLS-miyabe

Sep 2018, accepted

11 Mar 2018, submitted to a Journal

**Title**

Muchnik degrees and Medvedev degrees of the randomness notions

**Type**

Research paper

**Publication**

The joint proceedings for ALC2015 and ALC2017 that will be published via World Scientific

**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

13 Sep, 2018 The slide file was uploaded.

**Title**

Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm

**Type**

A talk in CTFM2018

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CTFM-EFKP

13 Sep, 2018 The slide file was uploaded.

**Title**

A tutorial in game-theoretic probability and algorithmic randomness

**Type**

A talk in CTFM2018

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CTFM-tutorial

1 Feb 2016, accepted by TOCS

**Title**

Coherence of reducibilities with randomness notions

**Type**

Full paper

**Journal**

Theory of Computing Systems – October 2018, Volume 62, Issue 7, pp 1599–1619

DOI: https://doi.org/10.1007/s00224-017-9752-2

**Abstract**

Loosely speaking, when $A$ is “more random” than $B$ and $B$ is “random”,

then $A$ should be random.

The theory of algorithmic randomness has some formulations of “random” sets

and “more random” sets.

In this paper, we study which pairs $(R,r)$ of randomness notions $R$

and reducibilities $r$ have the follwing property:

if $A$ is $r$-reducible to $B$ and $A$ is $R$-random,

then $B$ should be $R$-random.

The answer depends on the notions $R$ and $r$.

The implications hold for most pairs, but not for some.

We also give characterizations of $n$-randomness via complexity.

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preprint

31 July, 2018 The slide file was uploaded.

**Title**

The law of the iterated logarithm

**Type**

Seminar in the lab

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LIL

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