Stochastic processes determined by a general success-breeds-success principle

The general “success-breeds-success” (SBS) principle as introduced in a previous paper extends the classical SBS principle in that the allocation of items over sources is determined by a more general rule than in the classical case. In this article we study the time evolution of the total number of sources, the average number of items per source and the number of sources with n items at time t, in the general SBS framework. Conditional as well as absolute expectations are calculated. Moreover, we investigate if and when these processes are martingales, supermartingales or submartingales. Stability results for the stochastic processes are obtained in the sense that we are able to determine when these processes converge. The article also studies the evolution of the expected average number of items per source.