  Consider a finite Abelian group (G,+), with |G|=p^r, p a prime number, and F:
G^N -> G^N the cellular automaton given by {F(x)}_n= A x_n + B x_{n+1} for any
n in N, where A and B 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 w=
(w_i:i<0) the Cesaro mean distribution of the time iterates of the automaton
with initial distribution P_w --the law P conditioned to w on the left of the
origin-- converges to the uniform product measure on G^N. The proof uses a
regeneration representation of P.
