Some theoretical properties of the geometric and α-series processes

The geometric process has been proposed as a simple model for use in reliability. Recently, the -series process was proposed as a complementary model which can be used in situations where the geometric process is inappropriate. In this article, we show that the increasing geometric process grows at most logarithmically in time while the decreasing geometric process is almost certain to have a time of explosion. The -series process grows either as a polynomial in time or exponentially in time. We also show that, unlike most renewal processes, the geometric process does not satisfy a central limit theorem, while the -series process does.