This is called a reinforced process. Simplest example is Polya's urn: Consider an urn with balls of two different colors. Each time we pick a ball, we put it back together with a new ball of the same color (hence this colors becomes more likely).
What you're asking for is quite vague. For any sequence of numbers in , you can define a process with values in such that, given , the probability of is where . If is increasing, this is a general reinforced process. The choice of this sequence is the only data you need.