Cascade channels with infinite memory

摘要

Two theorems are proved for a cascade channel with two components, one theorem regarding input memory and the other regarding output memory. First, we show that if both components are asymptotically input-memoryless, then the cascade channel is asymptotically input-memoryless as well. Further, we prove that if both components are α-mixing and additionally the second component is causal and asymptotically input-memoryless, then the cascade channel is α-mixing. The results allow to study memory properties of complex models by analyzing basic building blocks. Further, they can be applied to analyze memory properties of information sources at the output of a channel. The results are relevant, e. g., in connection with coding theorems, concentration inequalities, or central limit theorems. The considered model includes discrete- as well as continuous-time channels and sources with completely arbitrary alphabets.