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When a chain is irreducible (so each state can be reached from every other state, eventually), we quote that all states have the same character: all aperiodic / periodic with the same period, all transient , all recurrent , all null, all positive, or all ergodic.

Results of this type are called solidarity theorems.

I quote the above from lecture notes. However, I dont understand how can a Chain be irreducible, and have all states transient?

It really doesnt make sense to me. Can someone clarify this for me please ?

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Try the chain on the integer line with transition probabilities $p$ for transitions $x\to x+1$ and $1-p$ for transitions $x\to x-1$, when $p\ne\frac12$. –  Did Dec 12 '12 at 8:07
While we are at it, what happened to this question? –  Did Dec 28 '12 at 11:17

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