How to prove Cramér's theorem?

Question

Cramér's theorem:

Let $X,Y$ two independent random variables such that $X+Y$ is normal distributed, then $X$ and $Y$ are normal distributed.

I get the the original paper: Harald Cramér. "Über eine Eigenschaft der normalen Verteilungsfunktion". Mathematische Zeitschrift, 41 (1): 405–414, 1936. But I cannot understand German. Is there an English version?

Thanks a lot.

Answer 1

The proof can be found in "Random Variables and Probability Distributions", Harald Cramer, Cambridge Tracts in Mathematics and Mathematical Physics.

I can send you an extract if you want.

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