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!

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. 
