Find parameters of random vector

Question

Denote random vector $ {\displaystyle \mathbb {X}}= (X_1;X_2;...;X_n)^T$ ${\displaystyle \mathbb {X}}$ ~ $N_n(\mu,\Sigma)$.

Using definition of multivariable normal distribution, prove that random vector $ {\displaystyle \mathbb {Y}}=(X_1-X_2;X_2-X_3;...;X_{n-1}-X_n)^T$ has normal distribution and find his parameters.

Definition of normal radom vector is that the random vector $ {\displaystyle \mathbb {X}}=(X_1;X_2;...;X_n)^T$ is normal, when for any vector $ {h = (h_1;h_2;...h_n)^T}$ is $h^T{\displaystyle \mathbb {X}}$ normal random variable.

Thank you for any help.

Answer 1

Hint:

You can write $\mathbb Y=A\mathbb X$ for some matrix.

Then $h^T\mathbb Y=h^TA\mathbb X=g^T\mathbb X$ for $g=A^Th$, hence is a normal random variable for every $h$.

Allowing the conclusion that $\mathbb Y$ is normal.

On base of $\mathbb Y=A\mathbb X$ you can easily find expressions for mean and covariance.