Let (f,α) be the process given by an endomorphism f and by a finite partition $\alpha ={A_i}_{i=1}^s$ of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if $E(f,\alpha )\subset g:{\bigvee }_{{B_i}_{i=1}^s}suppg={\bigcup }_{i=1}^s{A}_i\times B_i$. We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the ’chequer-wise’ partition for σ × S, where σ is...