Consider a finite Abelian group $(G,+)$, with $|G|=p^r$, $p$ a prime number, and $\varphi: G^\mathbb{N} \to G^\mathbb{N}$ the cellular automaton given by $(\varphi x)_n=\mu x_n+\nu x_{n+1}$ for any $n\in \mathbb{N}$, where $\mu$ and $\nu$ are integers coprime to $p$. We prove that if $\mathbb{P}$ is a translation invariant probability measure on $G^\mathbb{Z}$ determining a chain with complete connections and summable decay of correlations, then for any ${\underline w}= (w_i:i<0)$the Cesàro mean distribution $${\cal M}_{\mathbb{P}_{\underline w}} =\lim_{M\to\infty} \frac{1}{M} \sum^{M-1}_{m=0}\mathbb{P}_{\underline w}\circ\varphi^{-m},$$ where $\mathbb{P}_{\underline w}$ is the measure induced by $\mathbb{P}$ on $G^\mathbb{N}$ conditioned by $\underline w$, exists and satisfies ${\cal M}_{\mathbb{P}_{\underline w}}=\lambda^\mathbb{N}$, the uniform product measure on $G^\mathbb{N}$. The proof uses a regeneration representation of $\mathbb{P}$.