I have a stationary ergodic process of random variables $\{X_n\}_{n\in\mathbb{N}}$ and a constant $r<1$. Suppose that $\mathbb{E}(\log^+|X_1|)<\infty$. Then the sum $$ \sum_{n=1}^{\infty} r^nX_n $$ converges, see for example lemma 2.1 of this link.
Question: Assume all the $X_n$ are absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$. Is the sum then absolutely continuous too? If not, are there extra assumptions that ensure this result?