Questions about stationary and limiting distributions of a discrete-time Markov chain

For a discrete-time Markov chain,

1. Is it right that there are no more than one limiting distribution, i.e., limiting distribution is unique if any?
2. If the chain has more than one recurrence irreducible classes, then is it right that the limiting distribution does not exist?
3. If the chain has only one recurrence class, does existence of transient states not affect the existence of a limiting distribution?
4. When both limiting and stationary distributions exist, is it possible there is some stationary distribution that is not a limiting distribution?
5. If interpreting the limiting distribution as limit of the distribution of $X_n$, in what sense does the sequence of distribution measures converge?

Thanks and regards!

-
-1: These look like exam questions, and I think you should take the effort to try to look up the answers in your textbook first. If you have specific questions about what you read, feel free to ask them here. –  Nate Eldredge Apr 30 '11 at 17:41
Maybe this math.stackexchange.com/questions/24759/… will be helpful. –  Byron Schmuland Apr 30 '11 at 17:59
@Byron: Thanks for the link! It is helpful indeed! I now have some new questions updated to replace the old ones. Would appreciate if you could read and/or reply. –  Tim May 1 '11 at 20:25
@Nate: Thanks! I have read several posts in this site and some theorems in some books, which basically answered my old questions. Now I have some new questions and just replaced my old ones with them. Would appreciate if you could read and/or reply. –  Tim May 1 '11 at 20:25