Constructing Neg Independent RV from another?

Question

If we have a sequence of independent random variables $X_{i}$ how can we construct a new sequence $Y_i=-X_i$ such that they have the same distribution and are independent of each other?

Answer 1

This is a nit-picky answer: it depends on what your underlying probability space is. I assume all your random variables are defined on a probability space $(\Omega,\mathcal F, P)$. If (as a result of poor planning) it turns out that $\mathcal F=\sigma(X_1,X_2,\dots)$ you are out of luck: you have used up all your randomness: there are no non-constant random variables that are independent of the $X_i$.

In such a case you'd wish you had started with $\Omega= \Omega\times\Omega$ or even $\Omega' = \Omega^{\mathbb N}$ instead. Or you'd use some circumlocution like: If the $X_i$ are independent, there is another probability space, supporting random variables $X_i'$ whose joint distribution is the same as the $X_i$s, and also random variables $-Y_i'$ whose joint distribution is also the same as the $X_i$ and independent of the $X_n'$.