I am currently working on a Stochastic Process question for my Probability Course. I have hard time understanding how to show that a Bivariate Normal Distribution specifies a stochastic process. I am given:
$X_1\sim N(0,1)$ and $X_2\sim N(0,1).$
Then, $(X_1,X_2)\sim N\left(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1, 1/2 \\ 1/2, 1 \end{bmatrix} \right)$
As a result, my cdf is $f(x_1,x_2) = \frac{1}{(\pi \sqrt{3})} \exp( -\frac{2}{3}(x_1^2 - x_1x_2 + x_2^2))$
Does this specify a stochastic process with state space $S = R$, time domain $T = [1,2]$? Justify the answer.
From my lecture notes, I have the following:
i) $f(x) \geqslant 0$ almost everywhere
ii) Integral of $f(x)\,dx$ over $R^k$ is equal to 1.
If $f$ satisfies i and ii above, then $f$ is a density. Then by Kolmogorov Consistency Theorem, we have that $f$ satisfies a stochastic process (everything marginalizes properly).
But then, where does $T = [1,2]$ come in?