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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.

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By stable, I assume you mean Levy-Stable (not to be confused with Levy-Processes)? If so, books on the subject are pretty limited. Here is the list that I have found so far:

Chap. 1 of Nolan's (not yet published) book.

"One-Dimensional Stable Distributions" by Zolotorev

"An Introduction to Probability Theory and Application, Vol. 2" by Feller has a couple of chapters and discussion on the subject of stable/semi-stable distributions.

"Stable Non-Gaussian Processes: Stochastic Models With Infinite Variance" by Samorodnitsky and Taqqu, though I have not read this myself.

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This looks like it might be up your alley.

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That is indeed quite a nice paper, thank you! – little_probabilist Aug 16 '10 at 17:17

Have you looked into Jacod & Shiryaev [JS] ? The book should contain what you need, but it is somewhat monstrous.

Another book on functional limit theorems based on martingale problems is by Ethier & Kurtz. It is a much more pleasant (and highly recommended) reading, although [JS] is more likely to have something related to your request.

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