The Quasi-equivalence between the Definitions of Partial Randomness

In the recent literature, many definitions of partial randomness of reals have been proposed and studied rather discretely. For instance, it is known that for a computable real epsi isin (0,1), strong Martin-Lof epsi-randomness is strictly stronger than Solovay epsi-randomness which is strictly stronger than weak Martin-Lof epsi-randomness. In the present work, we firstly give several new definitions of partial randomness - strong Kolmogorov epsi-randomness and weak/strong DH-Chaitin epsi-randomness. Then, we investigate the relation between epsi-randomness by one definition and epsi´-randomness by another. Finally, we show that all of the known definitions of epsi-randomness are quasi-equivalent.