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Can some one help me point in the right direction or point to some resources that will help me prove that sum of two jointly distributed Gaussian r.v. with a given correlation coefficient is also a Gaussian r.v. Thanks

user100503
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    Many people take the _definition_ of jointly Gaussian random variables as a collection of random variables such that $\sum_i a_iX_i$ is a Gaussian random variable for _all_ choices of real numbers $a_i$. Thus, what you are asking for is a tautology for these people. Since $$\begin{align}E[X+Y]&= E[X]+E[Y]\\\operatorname{var}(X+Y)&=\operatorname{var}(X)+\operatorname{var}(Y)+2{\rho}\sqrt{\operatorname{var}(X)\operatorname{var}(Y)}\end{align}$$ apply even for _nonGaussian_ random variables, there is little left to do... – Dilip Sarwate Apr 12 '15 at 22:55
  • Thank you. But is the proof this simple? Don't I have to show something more? – user100503 Apr 12 '15 at 23:04
  • If I recall correctly, it was shown in Kendall's *The Advanced Theory of Statistics. Vol. 1: Distribution Theory*. – corey979 Apr 13 '15 at 00:02
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    What is _your_ definition of jointly Gaussian random variables? If you use the one stated in my previous comment, then you don't need to "show" very much more than what I wrote. If you start from the joint density, then there can be a lot of algebra involved. A proof starting from characteristic functions, instead of joint densities, is given in [this answer](http://stats.stackexchange.com/a/19953/6633) written by @probabilityislogic – Dilip Sarwate Apr 13 '15 at 13:14

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