Basic finite dimensional distribution question I'm having trouble wrapping my head around the basic idea of a finite dimensional distribution.  Suppose $(\Omega, \Bbb P, \mathcal{F})$ is a probability space. 
Let $(X_{t})_{t \geq 0}$ be a stochastic process, i.e., $X_{t}: \Omega \to \Bbb R $ is a random variable for each $t$.  
Let $\mu_{t}$ be the distribution measure of $X_{t}$ on $\mathcal{B}(\Bbb R)$, i.e., $\mu_{t}(A) = \Bbb P(X^{-1}(A))$ for each $A \in \mathcal{B}(\Bbb R)$.
Given $t_{1}, t_{2}, \dots, t_{n} \geq 0$, we can talk about the finite dimensional distribution of the vector $(X_{t_{1}},X_{t_{2}},\dots,X_{t_{n}})$.  Specifically, this is the measure $\mu$ defined by:
$$\mu(D) = \Bbb P \{ \omega \in \Omega \mid (X_{t_{1}}(\omega), X_{t_{2}}(\omega), \dots, X_{t_{n}}(\omega)) \in D \} $$ where $D \subseteq \Bbb R^{n}$ is a Borel set.  So $\mu$ is a measure on $\mathcal{B}(\Bbb R^{n})$.
For some reason, I'm having trouble understanding what this measure is doing.  For example, what if we have a measurable rectangle $A_{1} \times \cdots \times A_{n}$?  My gut is telling me that the measure of this is just the product of the distribution measures, i.e., $\mu(A_{1} \times \cdots \times A_{n}) = \mu_{t_{1}}(A_{1})\mu_{t_{2}}(A_{2}) \cdots \mu_{t_{n}}(A_{n})$.  Is that right?  Can anyone give me more information about this distribution?  I'm very new to this material.
 A: A finite-dimensional distribution is just the joint distribution of a random vector. You pick various time points $t_1,t_2,\ldots,t_n$, observe the stochastic process at those time points, and ask yourself "what's the distribution of this random vector?"
To define a FDD it is sufficient to specify the joint distribution on rectangles $A_1\times A_2\times \cdots\times A_n$ (as you've suggested), or  on products of half-infinte intervals $(-\infty,a_1]\times(-\infty,a_2]\times\cdots\times(-\infty,a_n]$ (to generalize the one-dimensional CDF of a scalar random variable). In general, these joint distributions won't correspond to product measure unless the selected time points yield independent variables; typically you get a joint distribution of correlated random variables. For example, if the stochastic process is Brownian motion, then the finite-dimensional distributions will be correlated multivariate gaussian.
The simplest finite-dimensional distribution occurs when $n=1$: you get just the distribution of $X_t$, the process observed at a single time point.
