Definition of two dimensional Gaussian process

If $$X=(X_t)_{t \geq 0}$$ is a one-dimensional Gaussian real process I know that it means that $$(X_{t_1}, X_{t_2}, \ldots , X_{t_n})$$ has $$n$$-dimensional Gaussian distribution for any $$0 \leq t_i< t_{i+1}$$ for $$i=1,2,....,n-1$$ and $$n \geq 1$$. What if $$(X,Y)$$ is a two-dimensional Gaussian process. Does it mean that $$(X_{t_1}, X_{t_2}, \ldots , X_{t_n}, Y_{t_1}, Y_{t_2}, \ldots , Y_{t_n})$$ has $$2n$$-dimensional Gaussian distribution for any $$0 \leq t_i< t_{i+1}$$ for $$i=1,2,....,n-1$$ and $$n \geq 1$$?

Note An answer is given in the post Definition of a $\mathbb{R}^d$-valued Gaussian process.

• I think a two-dimensional Gaussian process is more commonly understood to be a process where the index set is two-dimensional. This is contrasted with your one-dimensional process indexed by $\mathbb{R}$. May 3, 2021 at 14:18
• I don't agree. Gaussian processes on higher dimensional index sets are rather referred to as Gaussian fields. An $n$-dimensional Gaussian process, clearly is a Gaussian process valued $\mathbb{R}^{n}$. May 3, 2021 at 17:15
• Now in order to also provide a hint towards a possible answer. One way to characterise a multivariate Gaussian RV $X$ is by imposing that the inner product $a\cdot X$ shall be a (scalar) Gaussian for every $a\in\mathbb{R}^{n}$. Can you cook up an answer yourself now? May 3, 2021 at 17:17
• @Tobsn Thank you. I know that definition of multivariate Gaussian random variables. I think I found an answer to my question in a related post, math.stackexchange.com/questions/3313305/… May 5, 2021 at 5:39
• It's not a good answer. I may repeat myself: try to cook up a reasonable definition using the same philosophy as used in the definition of a multivariate Gaussian RV. May 5, 2021 at 6:03