Let's suppose I have a countable state discrete time MC that is known to be transient, irreducible and reversible with respect to some measure that assigns positive finite mass to each singleton, but is not necessarily a finite measure. Fix a state $a$ and let $x$ vary. Is it necessarily true that $P^a(\text{Chain hits x at some time})\rightarrow 0$ as $x\rightarrow\infty$? What about $P^x(\text{Chain hits a at some time})$?
2026-04-02 15:25:41.1775143541
Transience question for Markov Chains
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