Suppose that $\{X_k:k\in\mathbb Z\}$ is a linear process, i.e. a sequence of random variables such that $$ X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j} $$ for each $k\in\mathbb Z$, where $\{\psi_j:j\ge0\}$ is a real sequence such that $\sum_{j=0}^\infty|\psi_j|<\infty$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed random variables with $\operatorname E|\varepsilon_0|<\infty$.

Is the linear process $\{X_k:k\in\mathbb Z\}$ ergodic for the mean, i.e. does the average $n^{-1}\sum_{k=1}^nX_k$ converge almost surely (or in probability) to $\operatorname EX_0$ as $n\to\infty$?

If we assume that $\operatorname E\varepsilon_0^2<\infty$, then $\sum_{j=0}^\infty|\psi_j|<\infty$ implies the ergodicity for the mean (see p. 52 of Time Series Analysis by James D. Hamilton), but I'm interested in the case when we only assume that $\operatorname E|\varepsilon_0|<\infty$.

Any help is much appreciated!