We consider a class of so-called cesaro sequences over a finite alphabet, which possess the property of stability of the relative frequen- cies of occurrence of an arbitrary word in growing initial segments. This class includes a class of periodic sequences. A finite state automaton is called cesaro-hereditary if it processes cesaro sequences in the input alphabet into cesaro sequences in the output alphabet. Examples of automata that possess and do not possess the property of cesaro-heredity are given. The conditions are proposed so that the automaton possesses and does not possess the property of cesaro-heredity. © Springer Nature Switzerland AG 2020.