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 1


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.


$X \sim N_p(\mu,\Sigma)$. 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 $X \sim N_p(\mu, \Sigma )$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.