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This might be a naive question, but suppose I have two random variables f and g that are dependent, and I have a third random variables k that is independent of both. When can I say that f and g + k are independent random variables? If that's not true, then what is one way to turn two dependent variables into independent ones (by some non-trivial operation on one of them, say)?

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  • $\begingroup$ Intuitively, I think it's unlikely for $f$ and $g+k$ to be independent (if we're talking real-valued random variables), since the value of $f$ influences the value of $g$ and hence the maximum (say) value of $g+k$. For example, take $f\in\{-1,1\}$ with equal probability, take $g=f$, and take $k$ uniformly distributed on $(-1,1)$. One might be able to cook up exceptions, though. $\endgroup$ – Greg Martin Nov 19 '13 at 4:45
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  1. f and g + k will never be independent.
  2. I'm not aware of any general method to make T(f) independent of g where T is some but function (possibly stochastic as well).
  3. But if you replaced "independent" by "non-correlated" then there may be ways. Not sure but check/google "method of instrumental variables"
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