Markov Chains: Periodicity and Ergodicity

Question

I found the dynamical system definition of ergodicity to be very intuitive:

$T$, a measure preserving transformation wrt $(\Omega,\mu)$ is ergodic wrt $\mu$ if for all $A \subset \Omega$, $T^{-1}A = A$ means $\mu(A) = \{0,1\}$.

When I approach Markov chains, my reading is that we have to consider the measure space for all possible sequences, under measure from Kolmogorov's Extension,

$$(\Omega^{\mathbb{N}}, \mu^{\mathbb{N}})$$

And take transformation $T$ to be a shift of a single step, ie.

$$T(x_1,x_2,x_3,\ldots) = (x_2,x_3,\ldots)$$

But lots of definitions for ergodic markov chains (including the one on Wikipedia) uses aperiodicity. Where does periodicity come in? Is aperiodicity a necessary and sufficient condition for the Markov Chain to be ergodic? Can't a Markov chain be periodic and ergodic? Eg. Consider the 2 state system $p_x(y) = 1, p_y(x)=1$

Answer 1

As understood in our book, ergodicity means convergence to the stationary distribution of the Markov chain irrespective of the initial condition or distribution. Therefore, if a Markov chain is periodic, it cannot be ergodic because it does not converge in distribution to a limit. That is, there is no limit to the sequence $$\vert\vert K^n(x,\cdot)-\pi\vert\vert_{\text{TV}}$$ where $K$ denotes the Markov kernel, $\pi$ the stationary distribution, and $||\cdot||_{\text{TV}}$ the total variation norm. I suggest you read the easily accessible Markov Chains and Stochastic Stability by Seyn Meyn and Richard Tweedie.

There exist however ergodic theorems in Markov chain theory that do not require periodicity, like Birkhoff-Kinchin's theorem, the Markov equivalent of the Law of Large Numbers.