How is a Gaussian random process different from a Gaussian random variable?

Question

I am from a physics background with lousy mathematical training. I am trying to delve into the subject of stochastic processes as applied to physics. I have a pretty naive question.

A random variable $x$ ($-\infty<x<+\infty$) for which the normalized probability density function is $$p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big(-\frac{(x-\mu)^2}{2\sigma^2}\Big)$$ with $\mu$ being any real number and $\sigma$ being a positive number, is called a Gaussian random variable (GRV).

Question $1$ How is a GRV different from and related to a continuous Gaussian random process $x(t)$ where $t$ is a continuous parameter (e.g., time)? Should I think of Gaussian random process $x(t)$ as a non-denumerably infinite number of GRVs corresponding to each value of $t$?

Question $2$ A multivariate Gaussian distribution is of a set of $n$ GRVs (${\bf x}=(x_1,x_2,...x_n)$) is given by the normalized probability density function $$p({\bf x})=\frac{1}{\sqrt{(2\pi)^n\det({\boldsymbol \sigma})}}\exp\Big[-\frac{1}{2}({\bf x}-{\boldsymbol \mu})^T{\boldsymbol\sigma}^{-1}({\bf x}-{\boldsymbol\mu})\Big]$$ where ${\boldsymbol\mu}$ is a column vector of the mean values i.e. $\mu_i=\langle x_i\rangle$ and ${\boldsymbol\sigma}$ is a symmetric nonsingular matrix with nonnegative real elements, called the covariance matrix. If the answer to $(1)$ is 'yes', how is a multivariate Gaussian different from Gaussian random process if we can identity $x_1$ as $x(t_1)$, $x_2$ as $x(t_2)$... etc?

Answer 1

To add a little to T_M's answer,

Should I think of Gaussian random process 𝑥(𝑡) as a non-denumerably infinite number of GRVs corresponding to each value of 𝑡?

Yes, but there is a little more to it than just that.

You need the property that any finite set of two or more random variables from a Gaussian process enjoys a multivariate Gaussian distribution. As T_M points out, it cannot said that a Gaussian random process is the same as a multivariate Gaussian distribution; they are different. However, the requirement about multivariate Gaussianity is very important. Note that a finite collection of (two or more) Gaussian random variables need not necessarily have a multivariate Gaussian distribution. For example, if $\phi(x)$ is the standard Gaussian density function, then $$f_{X,Y}(x,y) = \begin{cases} 2\phi(x)\phi(y), & \text{if}~ x \geq 0, y \geq 0, ~\text{or if}~ x < 0, y < 0,\\ 0,& \text{otherwise}.\end{cases}$$ is a joint density function of two standard Gaussian random variables $X$ and $Y$ but is not a bivariate Gaussian density function. Bivariate Gaussian densities are nonzero everywhere in the plane, but the density above has value $0$ in the second and fourth quadrants.

In short, you need to add to your thinking the idea that "a Gaussian process is a non-denumerably infinite number of GRVs" the additional requirement that every finite subset enjoys a multivariate Gaussian distribution.

Answer 2

Should I think of Gaussian random process 𝑥(𝑡) as a non-denumerably infinite number of GRVs corresponding to each value of 𝑡?

Yes

In other words, is a Gaussian random process same as a multivariate Gaussian distribution?

A multivariate Gaussian is not the same as a non-denumerably infinite number of GRVs, so this is not a rephrasing of the previous question. The answer here is 'no'.