This note is concerned with asymptotic stability analysis of discrete-time linear time-varying (DLTV) systems. The contribution can be summarized as follows. First, the difference between nonuniformly exponential stability (ES), which was not well recognized in the literature, and uniformly ES (UES) was revealed and illustrated by numerical examples. Second, some state transition matrices and difference Lyapunov inequalities (DLIs) based necessary and sufficient conditions were derived for testing different stability concepts including asymptotic stability (AS), ES, and UES of general DLTV systems. The main feature of the DLIs-based criteria is that the time-difference of the Lyapunov function is not required to be negative definite. Finally, the ES concept was utilized for analysis stability of a class of upper triangular DLTV systems and perturbated DLTV systems. The effectiveness of the proposed results was demonstrated by some examples.