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Unpublished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.