It is shown that the gamma distribution with shape parameter α can be obtained through a p-thinning for every 0<p<1, when 0<α•1. In the case α>1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0<α•1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.
