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

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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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.

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