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In the stationary case, I know that if the chain is irreducible and aperiodic, it is Ergodic. But in the non-stationary case, i can not comprehend the content deeply. I want to know if Irreducibility holds for the Ergodic non-stationary Markov chain.

Generally, i want to know what are the main differences between Ergodicity of a stationary Markov chain and non-stationary one?

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Generally, i want to know what are the main differences between Ergodicity of a stationary Markov chain and non-stationary one?

This question could be a better question if formulated better.

(1)Stationary process

You are right about the fact that when we have a stationary Markov process then due to the discrete renewal theorem [1] we can see that there exists a stationary distribution for irreducible aperiodic chains and hence the Markov chain is ergodic. This is a special case of the Birkhoff's ergodic theorem which may apply to continuous time situation as well.

(2)From stationary to nonstationary process

It is well known that if the dependence between different states are not too strong then the Markov chain can possess ergodic properties as well[3]. The question is how to depict dependence between different states $X_s,X_{s+\Delta s}$. There are different ways like Bulinski's bounded Lipshcitz dependence[3], Kolmogorov's maximal correlation coefficient[5] and Rosenblatt's mixing coefficients[4]. The last one is becoming more and more popular set of conditions that are used through out physics, computer science and other subjects. If the nonstationary process $X_s$ satisfy some restrictions concerning about their mixing coefficients(i.e. dependence) Then the ergodicity still holds for such kind of stochastic processes as mentioned [5].

(3)'Wilder' nonstationary process

For example the sofic process will admit finite Markovian representations but themselves are not Markov chains of any finite order[5]. These processes will usually lead to chaotic behavior and therefore the ergodicity obviously collapse here and therefore we are unable to discuss ergodicity. In short the main difference between ergodicity of stationary and nonstationary processes is that whether their behavior can be regarded as some kind of orbit generated by transition kernel groups, which I think is the essenc ergodicity lies at.

Reference

[1] Karlin, Samuel. A first course in stochastic processes. Academic press, 2014.

[2]http://mathworld.wolfram.com/BirkhoffsErgodicTheorem.html

[3]Bulinskii, Aleksandr Vadimovich. Limit theorems for associated random fields and related systems. Vol. 10. World Scientific, 2007.

[4]Shields, Paul C. "The ergodic theory of discrete sample paths." Graduate Studies in Mathematics, American Mathematics Society (1996).

[5]Ibragimov, Il'dar Abdullovich. "A note on the central limit theorems for dependent random variables." Theory of Probability & Its Applications 20.1 (1975): 135-141.

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