**News**

Feb, 2011, Published

Dec 13, 2010, Submitted

**Title**

Generalizing randomness

**Type**

Survey

**Journal**

RIMS Kokyuroku

No. 1729

“Formal Systems and Computability Theory” (2010/09/13-2010/09-17)

pp. 84-94

**Download**

preprint

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# Category: Publication

## Generalizing randomness

## A Random Sequence of Reals

## An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

宮部賢志（ミヤベケンシ）

**News**

Feb, 2011, Published

Dec 13, 2010, Submitted

**Title**

Generalizing randomness

**Type**

Survey

**Journal**

RIMS Kokyuroku

No. 1729

“Formal Systems and Computability Theory” (2010/09/13-2010/09-17)

pp. 84-94

**Download**

preprint

Since this paper was rejected by a Journal, I will divide the results, extend them and publish by some papers.

**News**

Jan 11, 2011, Rejected

Aug 2, 2010, Submitted to a Journal

**Title**

A Random Sequence of Reals

**Type**

Fullpaper

**Journal**

Unpublished

**Download**

preprint

Japanese summary

**Abstract**

We define a random sequence of reals as a random point on a computable topological space. This randomness has three equivalent simple characterizations, namely, by tests, by martingales and by complexity. We prove that members of a random sequence are relatively random. Conversely a relatively random sequence of reals has a random sequence such that each corresponding member is Turing equivalent. Furthermore strong law of large numbers and the law of the iterated logarithm hold for each random sequence.

**Title**

An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

**Type**

Fullpaper

**Journal**

Notre Dame Journal of Formal Logic, 51(3):337-349, 2010.

Permanent link to this document in this Journal.

Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010

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preprint

Japanese summary

**Abstract**

Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considerint the join of infinitely many reals which are random relative to each other.

In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi’}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$. We also prove that there exists A such that, for each n, the real $\Omega^A_M$ is highn for some universal Turing machine M by using the extended van Lambalgen’s Theorem.

**Cited by**

@article{bienvenu2010ergodic,

title={{Ergodic-type characterizations of algorithmic randomness}},

author={Bienvenu, L. and Day, A. and Mezhirov, I. and Shen, A.},

journal={Programs, Proofs, Processes},

pages={49–58},

year={2010},

publisher={Springer}

}

@article{bienvenu2011constructive,

title={A constructive version of Birkhoffʼs ergodic theorem for Martin–L{\”o}f random points},

author={Bienvenu, L. and Day, A. and Hoyrup, M. and Mezhirov, I. and Shen, A.},

journal={Information and Computation},

year={2011},

publisher={Elsevier}

}

@article{yu2011characterizing,

title={Characterizing strong randomness via Martin-L{\”o}f randomness},

author={Yu, L.},

journal={Annals of Pure and Applied Logic},

year={2011},

publisher={Elsevier}

}