I came across a theorem when I study ergodicity. It is an application of Birkhoff's Ergodicity Theorem.
Theorem. Let $T:\Omega \to \Omega$ be a measurable map such that $P$ is invariant with respect to $T$. Then the following statements are equivalent:
(i) $P$ is ergodic;
(ii) For every $X\in L^2(\Omega)$,$$\lim_{n \to \infty}\operatorname{Var}\left[\frac{1}{n}\sum_{k=0}^{n-1}X\circ \theta^k\right]=0;$$
(iii) For every $X\in L^2(\Omega)$,$$\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1}\operatorname{Cov}\left[X\circ \theta^k,X\right]=0.$$
Unfortunately I am stuck at the implication (ii) $\Rightarrow$ (iii). How should I prove it? Thanks for any comments.
The proof of the implication is as follows:
We have $$\frac{1}{n}\sum_{k=0}^{n-1}\text{Cov}[X\circ\theta^k,X]=\text{Cov}\left[\frac{1}{n}\sum_{k=0}^{n-1}X\circ\theta^k,X\right]\leq\text{Var}\left[\frac{1}{n}\sum_{k=0}^{n-1}X\circ\theta^k\right]^{\frac{1}{2}}\text{Var}[X]^{\frac{1}{2}}.$$ By (ii) we get the desired limit.