Is the claim that functions of independent random variables are themselves independent, true?
I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of a normal distribution, but I have not been able to find justification for it. It seems that some authors take it as given but I am not certain that this is always the case.
The most general and abstract definition of independence makes this assertion trivial while supplying an important qualifying condition: that two random variables are independent means the sigma-algebras they generate are independent. Because the sigma-algebra generated by a measurable function of a sigma-algebra is a sub-algebra, a fortiori any measurable functions of those random variables have independent algebras, whence those functions are independent.
(When a function is not measurable, it usually does not create a new random variable, so the concept of independent wouldn't even apply.)
Let's unwrap the definitions to see how simple this is. Recall that a random variable $X$ is a real-valued function defined on the "sample space" $\Omega$ (the set of outcomes being studied via probability).
A random variable $X$ is studied by means of the probabilities that its value lies within various intervals of real numbers (or, more generally, sets constructed in simple ways out of intervals: these are the Borel measurable sets of real numbers).
Corresponding to any Borel measurable set $I$ is the event $X^{*}(I)$ consisting of all outcomes $\omega$ for which $X(\omega)$ lies in $I$.
The sigma-algebra generated by $X$ is determined by the collection of all such events.
The naive definition says two random variables $X$ and $Y$ are independent "when their probabilities multiply." That is, when $I$ is one Borel measurable set and $J$ is another, then
$\Pr(X(\omega)\in I\text{ and }Y(\omega)\in J) = \Pr(X(\omega)\in I)\Pr(Y(\omega)\in J).$
But in the language of events (and sigma algebras) that's the same as
$\Pr(\omega \in X^{*}(I)\text{ and }\omega \in Y^{*}(J)) = \Pr(\omega\in X^{*}(I))\Pr(\omega\in Y^{*}(J)).$
Consider now two functions $f, g:\mathbb{R}\to\mathbb{R}$ and suppose that $f \circ X$ and $g\circ Y$ are random variables. (The circle is functional composition: $(f\circ X)(\omega) = f(X(\omega))$. This is what it means for $f$ to be a "function of a random variable".) Notice--this is just elementary set theory--that
$$(f\circ X)^{*}(I) = X^{*}(f^{*}(I)).$$
In other words, every event generated by $f\circ X$ (which is on the left) is automatically an event generated by $X$ (as exhibited by the form of the right hand side). Therefore (5) automatically holds for $f\circ X$ and $g\circ Y$: there's nothing to check!
NB You may replace "real-valued" everywhere by "with values in $\mathbb{R}^d$" without needing to change anything else in any material way. This covers the case of vector-valued random variables.
Consider this "less advanced" proof:
Let $X:\Omega_X\to\mathbb{R}^n,Y:\Omega_Y\to\mathbb{R}^m,f:\mathbb{R}^n\to\mathbb{R}^k,g:\mathbb{R}^m\to\mathbb{R}^p$, where $X,Y$ are independent random variables and $f,g$ are measurable functions. Then: $$ P\{f(X)\leq x \text{ and } g(Y)\leq y\}\\ = P(\{f(X)\leq x\}\cap\{g(Y)\leq y\})\\ = P(\{X\in\{w\in\mathbb{R}^n:f(w)\leq x\}\}\cap\{Y\in\{w\in\mathbb{R}^m:g(w)\leq y\}\}). $$ Using independence of $X$ and $Y$, $$ P(\{X\in\{w\in\mathbb{R}^n:f(w)\leq x\}\}\cap\{Y\in\{w\in\mathbb{R}^m:g(w)\leq y\}\})\\ = P\{X\in\{w\in\mathbb{R}^n:f(w)\leq x\}\}\cdot P\{Y\in\{w\in\mathbb{R}^m:g(w)\leq y\}\}\\ = P\{f(X)\leq x\}\cdot P\{g(Y)\leq y\}. $$
The idea is to notice that the set $$ \{f(X)\leq x\}\equiv\{w\in\Omega_X:f(X(w))\leq x\}=\{X\in\{w\in\mathbb{R}^n:f(w)\leq x\}\}, $$ so properties that are valid for $X$ are extended to $f(X)$ and the same happens for $Y$.
Yes, $g(X)$ and $h(Y)$ are independent for any functions $g$ and $h$ so long as $X$ and $Y$ are independent. It's a very well known results, which is studied in probability theory courses. I'm sure you can find it in any standard text like Billingsley's.
Not as an alternative, but as an addition, to the previous brilliant answers: the independence of functions of independent random variables is, in fact, quite intuitive.
Usually, we think that $X$ and $Y$ being independent means that knowing the value of $X$ gives no information about the value of $Y$ and vice versa. This interpretation obviously implies that you can't somehow "squeeze" any information out by applying a function (or by any other means actually).
