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Kenshi Miyabe

Publications and source records attributed to Kenshi Miyabe.

10 recordsLinked to original sources
arXiv2026

Ball codes: A coding characterization of Hausdorff and packing dimensions

We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in \(\mathbb R^n\). A \emph{ball code} assigns names to closed balls of \(\mathbb R^n\): it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty \(E\subseteq\mathbb R^n\), the Hausdorff dimension of \(E\) is the minimum, over all ball codes, of the supremum over \(x\in E\) of the lower asymptotic rate at which \(x\) can be described by names of balls containing it; the packing dimension is obtained in the same way from the upper rates. The Hausdorff characterization is a Euclidean counterpart of Ryabko's coding theorem for combinatorial sources. We obtain the packing characterization from the standard characterization of packing dimension by upper modified box-counting dimension, without encoding \(\mathbb R^n\) into a sequence space. We then derive the point-to-set principle of J.~Lutz and N.~Lutz by replacing a minimizing ball code with a rational one and storing the resulting countable codebook in an oracle.

math.LO↗
arXiv2019

Solovay reduction and continuity

The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether there exists a reducibility concept that corresponds to Hölder continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via Hölder continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

math.LO↗
arXiv2016

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

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 self-normalized 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.

math.PR↗
arXiv2016

Using almost-everywhere theorems from analysis to study randomness

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 \ML\ (ML) randomness. We establish several equivalences. Given a ML-random real $z$, the additional randomness strengths needed for the following are equivalent. \n (1) all effectively closed classes containing $z$ have density $1$ at $z$. \n (2) all nondecreasing functions with uniformly left-c.e.\ increments are differentiable at $z$. \n (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 $Π^0_n$ and $Σ^1_1$ classes.

math.LO↗
arXiv2014

Derandomization in Game-theoretic Probability

We give a general method for constructing a deterministic strategy of Reality from a randomized strategy in game-theoretic probability. The construction can be seen as derandomization in game-theoretic probability.

math.PR↗
arXiv2013

Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness

We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's Theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's Theorem where each half is computably random uniformly relative to the other.

math.LO↗
arXiv2011

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

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.

math.PR↗