This is a fairly broad request for references. I've tried a few hours of googling, but the usual process of chasing names and references doesn't seem to be converging on any must-read books or obviously state-of-the art articles/surveys, and I was hoping an expert around here might point me in the right direction.

Are there any good/state-of-the-art references for invariance principles (also called functional limit theorems and possibly other names) which relate to convergence to stable (but not Gaussian) processes? I'm particularly interested in papers that provide some grounding relative to the 'standard' $\alpha = 2$ case, such as when something like the Skorohod embedding theorem can hold (I have seen it for $\alpha > 1$, but am not sure what happens below there), or how methods change in the 'nicest' categories, such as iid variables, martingale difference arrays, strongly mixing stationary processes, etc. At the moment, I care most about the asymmetric, $\alpha \leq 1$ case but would be grateful for any reading suggestions.