# Markov Chains: Periodicity and Ergodicity

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 $p_x(y) = 1, p_y(x)=1$

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 Volatility by Seyn Meyn and Richard Tweedie.
• In dynamical systems, the only sensible distribution is the one induced by the sequence $(X_0,T(X_0),\ldots)$ in terms of frequency, as there is no random structure in the sequence. It is purely deterministic and never "forgets" its starting point. – Xi'an Dec 26 '16 at 13:56