# Independent stochastic processes and independent random vectors

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

Two stochastic processes $${X(t)∣t∈T}\lbrace X(t)\mid t\in T \rbrace$$ and $${Y(t)∣t∈T}\lbrace Y(t)\mid t\in T \rbrace$$ are said to
be independent, if for any positive
integer $$n<∞n<\infty$$, and any sequence
$$t1,…,tn∈Tt_1,\ldots,t_n\in T$$, the random
vectors
$$\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))$$
and
$$\boldsymbol{Y}:=(Y(t_1),\ldots,Y(t_n))\boldsymbol{Y}:=(Y(t_1),\ldots,Y(t_n))$$
are independent.

I was wondering if according to the
definition, for any positive integer
$$n,m<\inftyn,m<\infty$$, and any sequence
$$t_1,\ldots,t_n\in Tt_1,\ldots,t_n\in T$$ and any
sequence $$s_1,\ldots,s_m\in Ts_1,\ldots,s_m\in T$$, the
random vectors
$$\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))\boldsymbol{X}:=(X(t_1),\ldots,X(t_n))$$
and
$$\boldsymbol{Y}:=(Y(s_1),\ldots,Y(s_m))\boldsymbol{Y}:=(Y(s_1),\ldots,Y(s_m))$$
are also independent?

2. Some related questions are for two
independent random vectors $$VV$$ and
$$WW$$:

• Will any subvector of $$VV$$ and any subvector of $$WW$$ (the two subvectors
do not necessarily have the same indices in the original random vectors) be independent?
• For any two subvectors $$V_1V_1$$ and $$V_2V_2$$ of $$VV$$ and any two subvectors
$$W_1W_1$$ and $$W_2W_2$$ of $$WW$$, will the conditional random vectors $$V_1|V_2V_1|V_2$$
and $$W_1|W_2W_1|W_2$$ also be independent?

Thanks and regards!

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.