Positive recurrent CTMC with a null recurrent embedded DTMC

[Doobious]

A continuous-time markov chain is recurrent if and only if the embedded discrete time markov chain is recurrent. But, (as stated in several sources without proof or example that I can find) if the CTMC is positive recurrent the DTMC may be null recurrent and vice versa. Can someone provide an example where this occurs?

Thanks.

 

[bpet]

Depends what "the" embedded DTMC means, but wouldn't the expected return time for the CTMC be less than for the DTMC?