It is shown that infinite divisibility of random variables, such as first passage times in a stochastic process, is often connected with the existence of an imbedded terminating renewal process. The idea is used to prove that for a continuous time Markov chain with two, three or four states all first passage times are infinitely divisible but for more than four states there are first passage times which are not infinitely divisible.
"Infinite Divisibility in Stochastic Processes." Ann. Probab. 7 (3) 406 - 417, June, 1979. https://doi.org/10.1214/aop/1176995042