In one proof, I encountered the following reasoning: $$P(T_1=n,T_2=m\mid X_0=j)=P(T_1=n\mid X_0=j)P(T_2=m\mid X_0=j)$$ Where $T$s are waiting times between returns to a state, $X_0$ is the state at time $0$.
The text then goes on saying "thus $T_1$ and $T_2$ are independent".
Now, in general, conditional independence does not imply independence, so why is it true in this case?
edit: assuming the chain actually enters the states, otherwise they might not be independent at all!