# Distribution of linear combination of Gaussian random variables

Suppose that $$x_1, x_2, \dots$$ and $$y_1, y_2, \dots$$ are i.i.d. standard Gaussian random variables. Define the following

$$z_n = \sum_{i=1}^n c_{ni} x_i + y_n$$

We know that $$z_n$$ is a zero-mean Gaussian random variable. For any $$n\geq 1$$, we can write

\begin{align} z_{1:n} = \begin{bmatrix} z_1\\ z_2\\ \vdots\\ z_n \end{bmatrix} = \begin{bmatrix} c_{11} & 0 & \dots & 0\\ c_{21} & c_{22} & \dots & 0\\ \vdots\\ c_{n1} & c_{n2} & \dots & c_{nn} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} + I \begin{bmatrix} y_1\\ y_2\\ \vdots\\ y_n \end{bmatrix} = C_n x_{1:n} + I y_{1:n} \end{align}

$$z_{1:n}$$ can be written as a linear combination of $$n$$ i.i.d. standard Gaussian random variables $$v_1, \dots, v_n$$

$$z_{1:n} = A_n v_{1:n},$$

where $$A_nA^\intercal_n = C_nC^\intercal_n + I$$. Intuitively, I expect $$A_n$$ to be lower triangular (similar to $$C_n$$) because of the causality in the problem. However, my simulations suggest that $$A_n$$ is not triangular. Could you please explain why my approach/intuition is wrong?

I figured out that $$A_n$$ is not unique. In code, I used eigenvalue decomposition to compute $$A_n$$ and got a non-triangular matrix. To get a lower triangular matrix, use Cholesky decomposition.