Probabilistic programming provides a language to specify complex statistical models. Some important models from nonparametric statistics are naturally expressed as programs which use some form of local state. A stochastic process can then be viewed as a stateful object, which can be sampled from, but also keeps track of its previous call history. A related approach uses stochastic memoization. These features are challenging to reason about semantically. I will discuss this, in the context of models of linear logic and game semantics.