An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals

Notre Dame Journal of Formal Logic 51 (3):337-349 (2010)
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 considering 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 $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem
Keywords van Lambalgen's Theorem   martingale   high   Omega operator
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DOI 10.1215/00294527-2010-020
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