Central limit theorem for linear stochastic evolutions

We consider a Markov chain with values in [0, infinity)(Zd). The Markov chain includes some interesting examples such as the oriented site percolation, the directed polymers in random environment, and a time discretization of the binary contact process. We prove a central limit theorem for "the spatial distribution of population" when d >= 3 and a certain square-integrability condition for the total population is satisfied. This extends a result known for the directed polymers in random environment to a large class of models.