Any discrete quantum process is represented by a sequence of quantum channels. In this talk, we will consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence, present a theorem which shows that the composition of such a sequence of channels converges exponentially fast to a rank-one (replacement) channel, and use this formalism to describe the thermodynamic limit of ergodic Matrix Product States. We derive formulas for the expectation value of a local observable and prove that the 2-point correlations of local observables decay exponentially. We then analytically compute the entanglement spectrum across any cut (Joint work with Ramis Movassagh, to appear in Phys Rev X 2021).