It is proved that the class of stable probability measures and the class of self-decomposible probability measures on Banach spaces are closed under weak convergence.
"A Note on the Convergence of Stable and Class L Probability Measures on Banach Spaces." Ann. Probab. 1 (4) 716 - 718, August, 1973. https://doi.org/10.1214/aop/1176996900