Definition of a $\mathbb{R}^d$-valued Gaussian process

Question

Consider the following definitions from my lecture notes:

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A $\mathbb{R}^d$-valued random variable $X =(X_1, \dots, X_d)$ is called $d$-dim Gaussian if every linear combination of its coordinates is (one dimensional) normal distributed.

A $\mathbb{R}^d$-valued stochastic process $(X_t)$ is called Gaussian if all its finite dimensional distributions are Gaussian.

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I have some trouble with the dimensions: So to show that a $\mathbb{R}^d$-valued process $(X_t)$ is Gaussian, I pick some arbitrary times $t_1, \dots, t_n$ and consider the $(d \times n)$-valued (?) random variable $(X_{t_1}, \dots , X_{t_n})$. But then a linear combination of $(X_{t_1}, \dots , X_{t_n})$ won't be one dimensional?! So I can not use the given definition. Where is my mistake?

Answer 1

Yes, the definition is a little misleading. You have to think of your $d\times n$-valued random variable as a vector of length $d\cdot n$, not of a matrix, so just write all columns of the matrix one below the other.

Answer 2

I'm not sure if the definition is exact or where you'd like to use it. But by marginalization property of Gaussian random vectors, coordinates of $X$ are also Gaussian random variables (one dimensional). Also, Gaussian random variables remain Gaussian under linear (affine) transformations. So any linear combinations of coordinates of $X$ will be Gaussian. Hope this helps.