Understanding the Gaussianity Under Linear Transformations

Question

Problem

Time-series model:

$$\begin{align*} x_{t+1} = Ax_t + w, w \sim N(0,Q) \newline y_{t} = Cx_{t} + v, v \sim N(0, R) \end{align*}$$

where $w,v$ are i.i.d. Guassian noise variable, assume that $p(x_0)= N (\mu_0, \Sigma_0)$, What is the form of $p(x_0,x_1,...,x_T)$?

The solution provided by author is :

The joint distribution is Gaussian: $p(x_0)$ is Gaussian, and $x_{t+1}$ is a linear/affine transformations of $x_t$. Since affine transformations leave the Gaussianity of the random variable invariant, the joint distribution must be Gaussian.

My question

I don't understand what doesn this means:"Since affine transformations leave the Gaussianity of the random variable invariant, the joint distribution must be Gaussian."

• Can I think of joint distribution $p(x_0,x_1,...x_T)$ as green circle, and if we stretch it, the distribution still the same?
• What is random variable invariant?

Answer 1

Your questions touch the domain of discrete stochastic linear dynamical systems. (Partial) answer for your questions are:

1. A gaussian stochastic multivariable $z \in \mathcal{N}(\mu, \Sigma)$ has a mean vector $\mu \in \mathbb{R}^n$ and a positive covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$, as you picture.
2. Although the variables $v$ and $w$ are stochastic, their models do not change along variables t i.e. invariant.

I try to predict your future questions: The solution for time series $x_{t+1}$ as you define is given by expression below:

$$x_t = A^t x_0 + \sum\limits_{i=0}^{t-1} A^i w$$