# 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!

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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.