A central limit theorem for sums of dependent random variables

Question

While looking for central limit theorem for sums of dependent variables I have been suggested to check for this theorem by C. C. Heyde

My problem is, since the sequence $X_j$ with $E[X_0]=0$ is assumed to be stationary it should be that $E[X_j]=0$ for all $j$, so conditions (1) and (2) should be straightforwardly verified, where am I wrong?

Answer 1

The problem is that in the series, you do not have $\mathbb E\left[X_j\right]$ but $\mathbb E\left[X_jY\right]$ where $Y$ is some $\mathcal M_0$-mesurable random variable, hence we do not necessarily have that $\mathbb E\left[X_j\mathbb E\left[ X_N\mid\mathcal F_0\right]\right]=0$.

For example, we can consider a causal linear process $X_j=\sum_{i\geqslant 0}2^{-i}\varepsilon_i$, where $\left(\varepsilon_i\right)_{i\in\mathbb Z}$ is a zero-mean square-integrable i.i.d. sequence and $\mathcal M_0:=\sigma\left(\varepsilon_i,i\leqslant 0\right)$.