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)?
- f and g + k will never be independent.
- 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).
- But if you replaced "independent" by "non-correlated" then there may be ways. Not sure but check/google "method of instrumental variables"