Sum of Random Variables Jointly Gaussian

Question

Let $X_1$, ..., $X_n$ be random variables and define

$Y_k = \sum_{i=1}^k X_i $, $\hspace{5mm} k= 1, ...,n$

Suppose that $Y_1, ...,Y_n$ are jointly Gaussian. Are $X_1, ..., X_n$ jointly Gaussian?

Answer 1

Perhaps it will help to see the following: $$ X_1 = Y_1\\ X_2 = Y_2 - Y_1\\ X_3 = Y_3 - Y_2 \\ \vdots \\ X_n = Y_n - Y_{n-1} $$ Hence $$ {\bf X} = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ -1 & 1 & 0 & \cdots & 0\\ 0 & -1 & 1 & \cdots & 0\\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & -1 & 1 \end{pmatrix} {\bf Y} $$ where ${\bf X} = (X_1, ..., X_n)^{T}$ and ${\bf Y} = (Y_1, ..., Y_n)^{T}$.