Show that if $(X_1,Y_1)$ and $(X_2,Y_2)$ are independent random vectors, then $(X_1,Y_1,g(X_1,Y_1))$ and $(X_2,Y_2,g(X_2,Y_2))$ are as well.

Question

I am not sure how to go about showing that $(X_1,Y_1)$ and $(X_2,Y_2)$ being iid implies that $(X_1,Y_1,g(X_1,Y_1))$ and $(X_2,Y_2,g(X_2,Y_2))$ are iid for any measurable function $g:R^2\rightarrow R$.

This question is connected to a homework problem in which I'm trying to use the multivariate central limit theorem and the delta method to obtain an asymptotic distribution for the sample covariance - it's an intermediate step.

Answer 1

Let $(\Omega, \mathcal{F},P)$ denote the underlying probability space and define $Z_1 = (X_1, Y_1)$ and $Z_2 = (X_2, Y_2)$. Pick $A_1, A_2 \in \mathcal{B}^2$ and $B_1, B_2 \in \mathcal{B}$ where $\mathcal{B}$ denotes the Borel sets. Then \begin{align} P(Z_1 \in A_1, g(Z_1) \in B_1,Z_2 \in A_2, g(Z_2) \in B_2) &= P(Z_1^{-1}(A_1 \cap g_1^{-1}(B_1)) \cap Z_2^{-1}(A_2 \cap g_1^{-1}(B_2)) \\ &=P(Z_1^{-1}(A_1 \cap g_1^{-1}(B_1))) P(Z_2^{-1}(A_2 \cap g_1^{-1}(B_2)) \\ &=P(Z_1 \in A_1, g_1(Z_1) \in B_1) P(Z_2 \in A_2, g_2(Z_2) \in B_2) \end{align} where the second equality follows because $Z_1$ and $Z_2$ are assumed independent.