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

News
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.

arXiv

## Randomness and Solovay degrees

News
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.

coromandel_JLA_final

## Computable Measure Theory

News
11 Mar 2018, Submitted

Title
Computable Measure Theory

Type
Survey in Japanese

Journal
RIMS Kokyuroku 25 – 27 Dec 2017

rims

## Null-additivity in the theory of algorithmic randomness

News
Gave up for publication.
Rejected.

Title
Null-additivity in the theory of algorithmic randomness

Type
Full paper

Journal

Abstract

## Using Almost-Everywhere Theorems from Analysis to Study Randomness

News
10 Oct 2016, published online
29 Feb 2016, Accepted by BSL
May 2015, Resubmitted.
3 Nov 2014. Submitted

Title
Using Almost-Everywhere Theorems from Analysis to Study Randomness
(with Jing Zhang and Andre Nies)

Type
Full paper

Journal
The Bulletin of Symbolic Logic, Volume 22, Issue 3
September 2016, pp. 305-331
arXiv

Abstract
We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin-Lo ̈f (ML) randomness. We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent.
(1) all effectively closed classes containing z have density 1 at z.
(2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.
(3) z is a Lebesgue point of each lower semicomputable integrable function.
We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff’s pointwise ergodic theorem. Lastly we study randomness notions for density of $\Pi^0_n$ and $\Sigma^1_1$ classes.