Showing two stochastic process are independent Let $X=\{X_t, t \ge 0\}$ and  $Y=\{Y_t, t \ge 0\}$ are two stochastic processes defined on common probability space. Then two show X and Y are independent, does it suffice to show for any $t$ and $s$, $X_t$ and $Y_s$ are independent? Or do we need stronger conditions?
You need to show that the sigma-algebras $\sigma(\{X_t \mid t \ge 0\})$ and $\sigma(\{Y_t \mid t \ge 0\})$ are independent, it isn't sufficient that all pairs $X_t, Y_t$ are independent.

This can easily be seen by the following example: Let $X, Y$ be two independent random variables with nontrivial distribution. Then the processes
$$\begin{align*}X_t &amp;= X \cdot I\{t = 0\} + Y \cdot I\{t = 1\} \\
Y_t &amp;= X \cdot I\{t = 1\} + Y \cdot I\{t = 0\}\end{align*}$$
are not independent, although all pairs $X_t, Y_t$ are independent.



Edit: The question was changed since I've answered it. To answer the new question, consider the standard example of random variables $X_1, X_2, X_3$ that are pairwise independent, but not jointly independent. Now consider the processes
$$\begin{align*}Y_t &amp;= X_1 \cdot I\{t = 0\} + X_2 \cdot I\{t = 1\} \\
Z_t &amp;= X_3 \cdot I\{t = 0\}\end{align*}.$$
You need to show that the sigma-algebras $\sigma(\{X_t \mid t \ge 0\})$ and $\sigma(\{Y_t \mid t \ge 0\})$ are independent, it isn't sufficient that all pairs $X_t, Y_t$ are independent.

This can easily be seen by the following example: Let $X, Y$ be two independent random variables with nontrivial distribution. Then the processes
$$\begin{align*}X_t &amp;= X \cdot I\{t = 0\} + Y \cdot I\{t = 1\} \\
Y_t &amp;= X \cdot I\{t = 1\} + Y \cdot I\{t = 0\}\end{align*}$$
are not independent, although all pairs $X_t, Y_t$ are independent.



Edit: The question was changed since I've answered it. To answer the new question, consider the standard example of random variables $X_1, X_2, X_3$ that are pairwise independent, but not jointly independent. Now consider the processes
$$\begin{align*}Y_t &amp;= X_1 \cdot I\{t = 0\} + X_2 \cdot I\{t = 1\} \\
Z_t &amp;= X_3 \cdot I\{t = 0\}\end{align*}.$$