Real life Cramer's Decomposition Theorem example.

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I just heard of CDT and "tried" to read online documents (Not to mention "understand" it).

Easy Version Definition:

Given independent normally distributed random variables ξ1, ξ2, their sum is normally distributed as well, and vice versa.

But all I can find is the "proves" of the theorem!

Could somebody be so kind and give me a real life example to help me understand it!?

Much appreciated!!!