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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?

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    I have suppressed the tag "statistics": this question is about probability. – 2017-01-16

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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.

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    thanks for the quick answer. But Im curious how the independence of "set of random variables" is defined. – 2017-01-16
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    https://en.wikipedia.org/wiki/Independence_(probability_theory)#More_than_two_random_variables – 2017-01-17
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    Helped a lot, thanks. – 2017-01-17