Independence of functions of random variables.

Question

My question is:

Let $X_1, X_2, ... , X_n$ be independent random variables. Are the functions of independent random variables, say $g_{1}(X_1,...,X_i), g_{2}(X_{i+1}, ... ,X_{j}) , ... , g_{l}(X_{m}, ..., X_{n})$ independent? Note that each function have different set of random variables as arguments. If not, under what restricted conditions might they be independent?

Answer 1

Fact: If $X_1,X_2,\ldots,X_n$ are independent random variables, then the random variables $g_1(X_1),g_2(X_2),\ldots,g_n(X_n)$ are also independent.

Now, think each "set of random variables" as a single random variable. So, you have $Y_1,Y_2,\ldots, Y_l$, all independent random variables. Thus, $g_i(Y_i)$ are also independent random variables.