Recently I learned that it is a widespread idea in applied math to approximate high rate Poisson processes by a Wiener process. I.e. take $N$ to be a homogeneous Poisson with rate $\lambda$, then for a large enough $\lambda$ one can select some time $T=T(\lambda)$ such that $N(t), 0\leq t\leq T$ is distributionally close to the constant-drift Wiener process on the same interval (the solution of the SDE $dX(t) = \lambda + \sqrt{\lambda} dW$ for white noise $dW$).
Of course stated like that it is not only non-rigorous but also not correct, but I guess that if ones takes right scaling (probably also $\lambda$-dependent), one can make this sort of statement more precise. E.g. one could compare $N(t)/\lambda$ and $dX(t) = 1 + 1/\sqrt{\lambda} dW$.
The question: What would be a good reference for the rigorous formulation of this kind of statement? I am particularly interested in where this approximation breaks down if one replaces homogeneous Poisson with an inhomogeneous one (and constant drift with a time-dependent one).
