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Let $\lbrace Xn \rbrace$, $n=0,1,..$ be a discrete time markov chain with one-step probability matrix $P = \lbrace P(i,j) \rbrace, (i,j) \in S$ and no absorbing state.

Let $$Q = \lbrace Q(i,j) \rbrace$$ $$Q(i,i) = 0$$ $$Q(i,j) = {P(i,j) \over (1 – P(i, i)) }$$ for $i \ne j$.

Let $ \lbrace Y_n \rbrace, n=0,1,…$ be a DTMC with one-step transition probability matrix $Q$.

How would I prove that $ \lbrace X_n\rbrace$ is recurrent/transient if and only if $ \lbrace Y_n \rbrace$ is recurrent/transient.

Initial Ideas: Intuitively I realize the chain $ \lbrace Y_n \rbrace$ contains the same exact transitions from state to state as $ \lbrace X_n\rbrace$ and hence recurrence/transience properties should not change from chain to chain ... but I am not able to translate this intuition into a formal proof.

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