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 $\epsilon\in(0,1)$, strong Martin-L$\ddot{o}$f $\epsilon$-randomness is strictly stronger than Solovay $\epsilon$-randomness which is strictly stronger than weak Martin-L$\ddot{o}$f $\epsilon$-randomness. In the present work, we firstly give several new definitions of partial randomness --- strong Kolmogorov $\epsilon$-randomness and weak/strong DH-Chaitin $\epsilon$-randomness. Then, we investigate the relation between $\epsilon$-randomness by one definition and $\epsilon'$-randomness by another. Finally, we show that all of the known definitions of $\epsilon$-randomness are quasi-equivalent.