Probabilistic programming languages aid developers performing Bayesian inference. These languages provide programming constructs and tools for probabilistic modeling and automated inference. Prior work introduced a probabilistic programming language, ProbZelus, to extend probabilistic programming functionality to unbounded streams of data. This work demonstrated that the delayed sampling inference algorithm could be extended to work in a streaming context. ProbZelus showed that while delayed sampling could be effectively deployed on some programs, depending on the probabilistic model under consideration, delayed sampling is not guaranteed to use a bounded amount of memory over the course of the execution of the program. In this paper, we the present conditions on a probabilistic program's execution under which delayed sampling will execute in bounded memory. The two conditions are dataflow properties of the core operations of delayed sampling: the m-consumed property and the unseparated paths property. A program executes in bounded memory under delayed sampling if, and only if, it satisfies the m-consumed and unseparated paths properties. We propose a static analysis that abstracts over these properties to soundly ensure that any program that passes the analysis satisfies these properties, and thus executes in bounded memory under delayed sampling.