### A tutorial in game-theoretic probability and algorithmic randomness

2018年9月13日　スライドアップロード

タイトル
A tutorial in game-theoretic probability and algorithmic randomness

CTFM2018の講演

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CTFM-tutorial

### Coherence of reducibilities with randomness notions

2017年2月1日　TOCS受理

タイトル
Coherence of reducibilities with randomness notions

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

### 重複対数の法則

2018年7月31日　スライドアップロード

タイトル

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LIL

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

2017年8月8日　オンライン
2017年7月28日　SPA受理

タイトル
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)

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

### Randomness and Solovay degrees

2018年3月19日 オンライン
2018年3月5日　JLA受理

タイトル
Randomness and Solovay degrees
(with A. Nies and F. Stephan)

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