A Japanese journal “Susemi”, which means a seminar of mathematics, features randomness titled “understanding randomness” on the February number in 2011.
Since this paper was rejected by a Journal, I will divide the results, extend them and publish by some papers.
Jan 11, 2011, Rejected
Aug 2, 2010, Submitted to a Journal
A Random Sequence of Reals
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.