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Alexandre Eremenko
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This is a very special case of the Darmois-Skitovich theorem, which deals with two arbitrary linear forms of the independent random variables. See, for example

Kagan, A.M., Linnik, Yu.V.; Rao, C.Radhakrishna Characterization problems in mathematical statistics,

or perhaps more accessible

Theory Probab. Appl., 57(3), 368–374. I. A. Ibragimov, On the Skitovich--Darmois--Ramachandran Theorem Theory Probab. Appl., 57(3), 368–374.

Or just type "Darmois-Skitovich theorem" on Google.

Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429