Dependence after absolutely continuous transformation

Question

Can you give me an example of a probability space $(\Omega,\mathcal{F},P)$ and a probability measure $Q\ll P$ as well as random variables $X$ and $Y$ on this space which are independent w.r.t. $P$ but dependent w.r.t. $Q$?

Answer 1

Let $X,Y$ be non-negative random variables on $(\Omega,\mathcal F,P)$ independent wrt $P$ and with $\mathbb E(X+Y)=1$.

Define $Q$ by stating that $Q(B)=\mathbb E(X+Y)1_B$ for $B\in\mathcal F$ so that $Q\ll P$.

Then:

• $\mathbb E_Q(XY)=\mathbb E(X+Y)XY=\mathbb EX^2\mathbb EY+\mathbb EX\mathbb EY^2$
• $\mathbb E_QX=\mathbb E(X+Y)X=\mathbb EX^2+\mathbb EX\mathbb EY$
• $\mathbb E_QY=\mathbb E(X+Y)Y=\mathbb EX\mathbb EY+\mathbb EY^2$

The expression that can be found on base of this for $\mathbb E_QX\mathbb E_QY$ is not the same as the expression for $\mathbb E_Q(XY)$.

There is choice for $\mathbb EX,\mathbb EX^2,\mathbb EY$ and $\mathbb EY^2$ and we can choose them on such a way that $\mathbb E_Q(XY)\neq\mathbb E_QX\mathbb E_QY$ which means that $X$ and $Y$ are not independent wrt $Q$.

It would not surprise me if someone comes up with a simpler example.