Convergence of ergodic averages

Let $$(X_n)_{n\in\mathbb N_0}$$ be a time-homogeneous Markov chain on a probability space $$(\Omega,\mathcal A,\operatorname P)$$ with transition kernel $$\pi$$, invariant measure $$\mu$$ and initial distribution $$\nu$$. Let $$f\in\mathcal L^1(\mu)$$ and $$A_{b,\:n}f:=\frac1n\sum_{i=b}^{b+n-1}f(X_i).$$

Assuming that the total variation distance $$|\mu-\nu\pi^n|$$ tends to $$0$$ as $$n\to\infty$$, it is claimed here in Theorem 3.18 that $$A_{b,\:n}f\xrightarrow{n\to\infty}\int f\:{\rm d}\mu.$$ In the proof, the author is basically reducing the problem to the case $$\nu=\mu$$ (i.e. the chain is started in stationarity). However, even in that case, we should need that $$\operatorname P_\nu:=\nu\pi$$ (composition of transition kernels) is ergodic with respect to the shift $$\tau:\mathbb R^{\mathbb N_0}\to\mathbb R^{\mathbb N_0}\;,\;\;\;(x_n)_{n\in\mathbb N_0}\mapsto(x_{n+1})_{n\in\mathbb N_0}.$$ What am I missing?

• Could you write explicitly what $\nu \pi$ is? I don't see why it's a measure on $\mathbb R^{\mathbb N_0}$. – Roberto Rastapopoulos Mar 2 at 17:03
• You could also add that the convergence seeked is in the almost sure sense. For the constant Markov chain ($X_{n+1} = X_{n})$, all measures are invariant and the statement obviously does not hold, so I agree that an additional ergodicity condition should be assumed. – Roberto Rastapopoulos Mar 2 at 17:08

If $$\mu$$ is only invariant and not ergodic,$$^\ast$$ if we take the simple case $$\nu=\mu$$, the ergodic averages $$A_{b,n}f$$ converge almost surely to the random variable $$\mathbb{E}_\mu[f|\mathcal{I}](X_k)$$ (which may alternatively be written as $$\mathbb{E}_\mathrm{P}[f(X_k)|X_k^{-1}\mathcal{I}]$$), where $$k$$ may be any natural number and $$\mathcal{I}$$ is the $$\sigma$$-algebra of measurable sets $$A$$ satisfying $$\pi(x,A)=1$$ for $$\mu$$-almost all $$x \in A$$. This limiting random variable is only equal (mod null sets) to the constant $$\mu(f)$$ if $$\mu$$ is ergodic.
So I think an additional condition is meant to be included in the theorem, namely that $$\mu$$ is ergodic.$$^\ast$$ Indeed, Corollary 2.15 -- which is probably the "stationary case" of the CLT referred to several times in Section 3.4.3 (including the statement of Theorem 3.18) -- explicitly assumes ergodicity.
[By the way, $$\mathrm{P}_\nu$$ is not $$\nu\pi$$ -- which would just be the same as $$\nu$$ if $$\nu=\mu$$ -- nor is it any other kind of "composition" of $$\nu$$ with $$\pi$$. It is a measure on the sequence space, just as you have written, defined as the law of a homogeneous Markov process with transition kernel $$\pi$$ and initial distribution $$\nu$$.]
$$^\ast$$Ergodicity of $$\mu$$ (with respect to $$\pi$$) is defined as meaning that the equivalent conditions in Theorem 2.1 are satisfied.