# Theory of the convergence of option prices using trees

My current understanding of the theory behind the convergence of options prices using trees is the following:

Suppose $$S = (S_{t})_{0\leq t\leq T}$$ is the underlying process and $$g(S_{t}:0\leq t\leq T)$$ is the option we want to evaluate. Then we want to construct trees $$S^{N}$$such that $$S^{N} \Rightarrow S$$ (Here $$\Rightarrow$$ denote weak convergence). Assume $$g$$ behaves well enough for simplicity.

1. If $$g$$ depends only on the terminal value $$S_{T}$$ and $$S$$ is lognormally distributed, then as in the CRR paper, we can construct $$S^{N}$$ by matching the mean and variance of each tree step and conclude $$S^{N}_{T} \Rightarrow S_{T}$$ by the classic central limit theorem.

2. If $$g$$ is path dependent, i.e: $$g$$ depends on the whole process $$(S_{t})$$, and $$S$$ is still lognormally distributed, then we might need to use some sort of "functional central limit", for example, Donsker’s Theorem, to conclude weak convergence of $$(S_{t}^{N})$$ to $$(S_{t})$$. I think this can be done by applying the same mean/variance matching for each step of the tree.

My question is (The trees can be non-recombining):

1. What is the most general class of processes we can approximate weakly using this "mean/variance matching of each step of the tree" construction?
2. For an arbitrary stochastic process, I think "mean/variance matching of each step of the tree" might not be sufficient for the tree to converge weakly to it, but is it a necessary condition?
3. Can you suggest some material regarding this part of the theory?

Any help is appreciated! Thank you.