# 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!

## 2 Answers

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 minor problem for the third question. What if they do not have densities? – Vim Jun 7 '17 at 4:28

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.