Cesaro-heredity property in the shift register family

Non-autonomous binary automata from three classes are considered: shift registers, generalized shift registers, shift registers with internal XOR. We study the cesaro-heredity property of automata from these classes, that is, their ability to inherit the property of stability of relative word frequencies in growing initial segments of the input sequence. It is shown that shift registers always have this property. Conditions are obtained under which generalized shift registers and shift registers with internal XOR do not have this property.