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I was wondering if the holding times are independent in a continuous-time Markov chain?

Similar question in a semi-Markov process?

From what I have read, it is not mentioned that the holding times are independent in both cases, but it is in Poisson processes and in renewal processes. So I would like to know if independent holding times are true in the two more general cases: a continuous-time Markov chain and a semi-Markov process, or if there are some famous counterexamples?

Thanks and regards!

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up vote 3 down vote accepted

No, the holding times are not necessarily independent, even for a continuous-time Markov chain.

Think about the following chain, with jump rates as shown:

Markov chain diagram

If $H_n$ is the $n$th holding time (so $X_s = X_{H_n}$ for $H_n \le s < H_{n+1}$), then $H_2$ and $H_3$ are not independent. You can do the computation explicitly if you like, but intuitively, if $H_2$ is large, you're much more likely to be in state $B$ than $E$. This means that your next jump will most likely be to $C$ rather than $F$, and so the next holding time $H_3$ is more likely to have rate $1$ than $100$, so $H_3$ is more likely to be large.

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Thanks! How did you draw that nice graph? –  Tim Apr 29 '11 at 19:38
    
Graphviz: graphviz.org –  Nate Eldredge Apr 29 '11 at 20:00
    
Thanks! For any two holding times, (1) if given their beginning states, will they be conditionally independent? (2) if given their beginning and end states, will they be conditionally independent? If yes to both (1) and (2), is it because of Markov property and how to explain that based on that? –  Tim May 9 '11 at 6:20
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