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==={{anchor|Transience}}{{anchor|Recurrence}}Transience and recurrence===
A state ''i'' is said to be transient if, given that we start in state ''i'', there is a non-zero probability that we will never return to ''i''. Formally, let the [[random variable]] ''T<sub>i</sub>'' be the first return time to state ''i'' (the "[[hitting time]]"):
:<math> T_i = \inf \{ n\ge1: X_n = i\}.</math>
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[[Kolmogorov's criterion]] gives a necessary and sufficient condition for a Markov chain to be reversible directly from the transition matrix probabilities. The criterion requires that the products of probabilities around every closed loop are the same in both directions around the loop.
Reversible Markov chains are common in [[Markov chain Monte Carlo]] (MCMC) approaches because the detailed balance equation for a desired distribution '''{{pi}}''' necessarily implies that the Markov chain has been constructed so that '''{{pi}}''' is a steady-state distribution. Even with time-inhomogeneous Markov chains, where multiple transition matrices are used, if each such transition matrix exhibits detailed balance with the desired '''{{pi}}''' distribution, this necessarily implies that '''{{pi}}''' is a steady-state distribution of the Markov chain.
==== Closest reversible Markov chain ====
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This Markov chain is not reversible. According to the [[Matrix norm#Frobenius norm | Frobenius Norm ]] the closest reversible Markov chain according to <math>\pi = \left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)</math> can be computed as
[[File:Mchain simple corrected C1.png|frameless|center]]
If we choose the [[probability vector]] randomly as <math>\pi=\left( \frac{1}{4}, \frac{1}{4}, \frac{1}{2} \right)</math>, then the closest reversible Markov chain according to the Frobenius norm is approximately given by
[[File:Mvchain approx C2.png|400px|frameless|center]]
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