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