Independent stochastic processes

Question

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks

Answer 1

In general, if $A,B:(\Omega,\mathcal{F},P)\to\mathbb{R}$ are independent random variables, then $A^2$ and $B^2$ also:

Like $f(x)=x^2$ is Borel measurable, then $A^2$ is $\sigma(A)$-measurable. Then $\sigma(A^2)\subseteq\sigma(A)$. By the same way, $\sigma(B^2)\subseteq\sigma(B)$.

So, If $C\in\sigma(A^2)$ and $D\in\sigma(B^2)$, then $C\in\sigma(A)$ and $D\in\sigma(B)$. Thus, using independence of $A$ and $B$, $P(C\cap D)=P(C)P(D)$ for all $C\in\sigma(A^2)$ and $D\in\sigma(B^2)$, which means, by definition, independence of $A^2$ and $B^2$.

Indeed, in general, as you can see, if $f,g:\mathbb{R}\to\mathbb{R}$ are Borel measurable functions and $A,B$ are independent, the argument is true and $f(A),g(B)$ are independents.