There is a small literature on these topics, mostly from the 90s. The names to look for are A. Jakubowski and M. Kobus (alone and together). For an example see Theorem 1.2 in http://www.sciencedirect.com/science/article/pii/S0047259X85710111. Unfortunately, I am not aware of neither a good general treatment not a textbook treatment. It is hard to believe that a very general theorem exists with convergence to stable limits because you need to control regular variation of the tails - a problem that does not quite occur in the Gaussian case.

There is a small literature on these topics, mostly from the 90s. The names to look for are A. Jakubowski and M. Kobus (alone and together). For an example see Theorem 1.2 in¹ https://www.sciencedirect.com/science/article/pii/S0047259X85710111. Unfortunately, I am not aware of neither a good general treatment not a textbook treatment. It is hard to believe that a very general theorem exists with convergence to stable limits because you need to control regular variation of the tails - a problem that does not quite occur in the Gaussian case.

¹M. Kobus, Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors, Journal of Multivariate Analysis, Volume 52, Issue 2, 1995, Pages 199-244, https://doi.org/10.1006/jmva.1995.1011