two irreducible chains on (0,1,2...)with transition probabilities p(ij) and q(ij). consider a new chain whose space is the union of the state spaces of the two chains except that zero states of both chains are glued together and after coming to zero the particle choose either first or second chain with probability 1/2.

if both first and second chain are positively recurrent then the new chain is positively recurrent??

if only the first is p.s. recu and second is null-recu, the new chain is null-recu??

so long....... any idea will be appreciate!!!!