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A state ''i'' is said to be essential or final if for all ''j'' such that ''i'' → ''j'' it is also true that ''j'' → ''i''. A state ''i'' is inessential if it is not essential.<ref>{{cite book|last=Asher Levin|first=David|title=Markov chains and mixing times|page=[https://archive.org/details/markovchainsmixi00levi_364/page/n31 16]|title-link= Markov Chains and Mixing Times |isbn=978-0-8218-4739-8|year=2009}}</ref> A state is final if and only if its communicating class is closed.
A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state.<ref name="PRS"/><ref name="
===Periodicity===
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:<math> p_{ii} = 1\text{ and }p_{ij} = 0\text{ for }i \not= j.</math>
If every state can reach an absorbing state, then the Markov chain is an [[absorbing Markov chain]].<ref name=
| first = Charles M.
| last = Grinstead
| first2 = J. Laurie
| last2 = Snell
| author-link2 = J. Laurie Snell
| title = Introduction to Probability
|date=July 1997
| publisher = American Mathematical Society
| isbn = 978-0-8218-0749-1
| chapter = Ch. 11: Markov Chains
| chapter-url = https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf}}</ref><ref name=Kem>
{{cite book
| first = John G.
| last = Kemeny
| author-link = John G. Kemeny
| first2 = J. Laurie
| last2 = Snell
| author-link2 = J. Laurie Snell
| editor-first = F. W.
| editor-last = Gehring
| editor2-first = P. R.
| editor2-last = Halmos
| title = Finite Markov Chains
| url = https://archive.org/details/finitemarkovchai00keme_792
| url-access = limited
| edition = Second
| orig-year = 1960
|date=July 1976
| publisher = Springer-Verlag
| ___location = New York Berlin Heidelberg Tokyo
| isbn = 978-0-387-90192-3
| pages = [https://archive.org/details/finitemarkovchai00keme_792/page/n235 224]
| chapter = Ch. 3: Absorbing Markov Chains
}}</ref>
===Reversible Markov chain{{Anchor|detailed balance}}===
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