Some results on effective randomness

We investigate the characterizations of effective randomness in terms of Martin-Lof tests and martingales. First, we address a question of Ambos-Spies and Kucera, who asked for a characterization of computable randomness in terms of tests. We argue that computable randomness can be characterized in terms of Martin-Lof tests and effective probability distributions on Cantor space. Second, we show that the class of Martin-Lof random sequences coincides with the class of sequences that are random with respect to computable martingale processes; the latter randomness notion was introduced by Hitchcock and Lutz. Third, we analyze the sequence of measures of the components of a universal Martin-Lof test. Kucera and Slaman showed that any component of a universal Martin-Lof test defines a class of Martin-Lof random measure. Further, since the sets in a Martin-Lof test are uniformly computably enumerable, so is the corresponding sequence of measures. We prove an exact converse and hence a characterization. For any uniformly computably enumerable sequence r(1), r(2),... of reals such that each r(n) is Martin-Lof random and less than 2(-n) there is a universal Martin-Lof test U-1, U-2,... such that U-n{0,1}(infinity) has measure r(n).