# Proof for linear combination of multivariate normal X?

Can anyone link to a proof for both parts of the statement below? I assume this question has been asked before but I wasn't quite sure what to search and couldn't find anything.

If $X$ is distributed as $N_p(\mu,\Sigma$) then any linear combination of variables $a'X = a_1X_1+a_2X_2+...+a_pX_p$ is distributed as $N(a'\mu,a'\Sigma a)$. Also,if $a'X$ is distributed as $N(a'\mu,a'\Sigma a)$ for every $a$, then $X$ must be $N_p(\mu,\Sigma$).

## 1 Answer

Your statement is similar to the statement of Cramér-Wald device. According to Cramér-Wald device "The distribution of $X_{p\times 1}$ is known iff the distribution of of $\alpha ' X$ is known $\forall \alpha \in \mathbb R^{p}$"

For proving use the concept of characteristic function. You can easily solve the problem on your own.

That's all from me.

Update

$X \sim N_p(\mu,\Sigma)$. So the characteristic function of $X$ is $\phi_X(t)=\exp(i t'\mu-\dfrac12t'\Sigma t)$.

Note that the characteristic function of $a'X$ is

\begin{align}\phi_{a'X}(t)&=E(\exp (it(a'X)))\\&=E(\exp (i(at)'X)))\\&=\exp(i (at)'\mu-\dfrac12(at)'\Sigma at)\\&=\exp(i t(a'\mu)-\dfrac12t(a'\Sigma a)t)\\&=\exp(i t(a'\mu)-\dfrac12t^2(a'\Sigma a))\end{align} By the uniqueness of characteristic function we can conclude $a'X \sim N(a'\mu, a'\Sigma a)$

\begin{align}\phi_{X}(\beta)&=E(\exp (i\beta'X))\\&=E(\exp (i(at)'X))\,[\text{consider} \,\beta =at]\\&=E(\exp (it(a'X)))\\&=\exp(i (at)'\mu-\dfrac12(at)'\Sigma at)\\&=\exp(i \beta'\mu-\dfrac12 \beta'\Sigma \beta)\,[\text{substitute} \,\beta =at]\end{align} Similarly by the uniqueness of characteristic function we can conclude $X \sim N_p(\mu, \Sigma )$