This article studies the stability problem of discrete-time switched positive linear systems (SPLSs) with marginally stable subsystems. Based on the weak common linear copositive Lyapunov function (weak CLCLF) approach, the switching property and the state component property are combined to ensure the asymptotic stability of SPLSs under three types of switching signals. First, considering the transfer-restricted switching signal described by the switching digraph, novel cycle-dependent joint path conditions are proposed in combination with state component digraphs. Second, under the time interval sequence, two types of path conditions are constructed for designing switching schemes. Third, necessary and sufficient conditions for the asymptotic stability of SPLSs under arbitrary switching are established. Finally, three examples are provided to illustrate the effectiveness of the proposed method.