# Independent stochastic processes and independent random vectors

1. 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?

2. 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!

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.
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.