Markov Processes with Infinitely Divisible Limit Distributions

To study special Markov processes with infinitely divisible limit distributions, the characteristic function was used. Some examples which have infinitely divisible limit distributions were given and a counterexample illustrated that not all the Markov processes had infinitely divisible limit distributions. An easily verified sufficient and essential condition which can assure Markov processes have limit distributions was given. Moreover a sufficient and essential condition ensuring that some special kind of Markov process has infinitely divisible limit distributions was given.