If a linear combination of variables is normally distributed, then the individual variables are also normally distributed

As far as I know this is a 'known' fact on Probability. In fact, it completely characterizes the normal distribution. No other distribution has this property.

My question is, do you happen to have a reference for this fact AND a reference for the proof of this fact. I can't find any.

Solution 1:

The fact: "The distribution of a $p$-dimensional random variable $\mathbf{U}$ is completely determined by the one-dimensional distributions of linear functions $\mathbf{T'U}$, for every fixed real vector $\mathbf{T}$", is a result due to Cramer and Wold, and its proof have the aid of characteristic function. Pls refer to C. R. Rao, Linear Statistical Inference and its Applications, 2nd Ed. John Wiley & Sons, Inc.(2002). p.517.