Markov processes with infinitely divisible limit distributions: some examples

A set of examples is described which suggests that members of a certain class of Markov processes have infinitely divisible limit distributions. Counter examples rule out such a possibility and raise the question of what further restrictions are required to guarantee infinitely divisible limits. Some related examples illustrate the same occurrence of infinitely divisible limit distributions. For both settings, an easily checked necessary and sufficient condition is obtained for the existence of a limit distribution.