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I just saw a theorem that sais that a continuous markov chain $(X_t)$ is recurrent if and only if it's jump chain is recurrent. I was wondering if it is the same thing for the concept of positive recurrence (finite expected return time). I would say intuitively that it is not the case but what example of chain can we find where :

  • The continous chain is positively recurrent but the jump chain not
  • The jump chain is positively recurrent but the continuous process not

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