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