Geodesic
Stability of the characterization of the multivariate normal distribution in the Skitovich--Darmois theorem
Zapiski Nauchnykh Seminarov POMI, Continuity and stability in the problems of probability theory and mathematical statistics, Tome 61 (1976), pp. 5-16

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A numerical estimate of stability is obtained in a multidimensional variant of the Skitovich–Darmois theorem when the original random vectors, as well as linear combinations of them, are assumed to be $\varepsilon$-independent. 
@article{ZNSL_1976_61_a0,
     author = {Yu. R. Gabovich},
     title = {Stability of the characterization of the multivariate normal distribution in the {Skitovich--Darmois} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--16},
     publisher = {mathdoc},
     volume = {61},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_61_a0/}
}
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Yu. R. Gabovich. Stability of the characterization of the multivariate normal distribution in the Skitovich--Darmois theorem. Zapiski Nauchnykh Seminarov POMI, Continuity and stability in the problems of probability theory and mathematical statistics, Tome 61 (1976), pp. 5-16. http://geodesic.mathdoc.fr/item/ZNSL_1976_61_a0/