Independent stochastic processes and independent random vectors 
*

*The definition for the two processes
to be independent is given by
PlanetMath:

Two stochastic processes $\lbrace X(t)\mid t\in T \rbrace$ and $\lbrace Y(t)\mid t\in T \rbrace$ are said to
be independent, if for any positive
integer $n<\infty$, and any sequence
$t_1,\ldots,t_n\in T$, the random
vectors
$\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))$
and
$\boldsymbol{Y}:=(Y(t_1),\ldots,Y(t_n))$
are independent.

I was wondering if according to the
definition, for any positive integer
$n,m<\infty$, and any sequence
$t_1,\ldots,t_n\in T$ and any
sequence $s_1,\ldots,s_m\in T$, the
random vectors
$\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))$
and
$\boldsymbol{Y}:=(Y(s_1),\ldots,Y(s_m))$
are also independent?


*Some related questions are for two
independent random vectors $V$ and
$W$:

*

*Will any subvector of $V$ and any subvector of $W$ (the two subvectors
do not necessarily have the same indices in the original random vectors) be independent?

*For any two subvectors $V_1$ and    $V_2$ of $V$ and any two subvectors
$W_1$ and $W_2$ of $W$, will the conditional random vectors $V_1|V_2$
and $W_1|W_2$ also be independent?



Thanks and regards!
 A: The answer to all your questions is yes. And they can be deduced from the following :
If two random vectors $\boldsymbol{X}:=(X_1, \ldots,X_n)$ 
and $\boldsymbol{Y}:=(Y_1, \ldots,Y_m)$ are independent, any pair of "marginalized" random vectors $\boldsymbol{X_A}$,  $\boldsymbol{Y_B}$ (each formed by arbitrary subsets of the originals) are independent.
This property (basically your second question) can be readily deduced from the definition of independence (factorization of joint densities) and marginalization. From this the other two follow.
A: Take $t_1, t_2, \ldots t_n, s_1,s_2, \ldots s_m \in T$ and define $X := (X(t_1), \ldots, X(s_m)), Y=(Y(t_1), \ldots , Y(s_m))$ . Now by definition $X$ and $Y$ are independent.Then take two projections $\pi_1 $ and $\pi_2$ from  $\mathbb{R}^{n+m}$  to $\mathbb{R}^{n}$  and $\mathbb{R}^{m} $respectively such that $\pi_1 \circ X =(X(t_1), \ldots, X(t_n))$ and $\pi_2 \circ Y =(Y(s_1), \ldots, Y(s_m)).$ Now as projections are continuous, both $(1)$ and $(2 )$ follows. 
