Implications of mutual independence between random variables Consider the random vectors $Z_1, Z_2, Z_3, Z_4$ and assume they are i.i.d.

Moreover, $Z_i\equiv (Y_i, X_i)$, where $Y_i$ $X_i$ are scalar random variables, and $\epsilon_i\equiv g(Y_i, X_i)$ for $i=1,2,3,4$ for some function $g$

I want to show that $\epsilon_i \perp X_i $ $\forall i$ implies $\epsilon_i \perp(X_1, X_2, X_3, X_4)$ $\forall i$

Could you help me? ($\perp$ denotes independence)



My attempt (correct?)


Let $i=1$
$Z_1, Z_2, Z_3, Z_4$ i.i.d. and $\epsilon_1\equiv f(Y_1, X_1)$ $\Rightarrow$ $(\epsilon_1, X_1) \perp (X_2, X_3, X_4) $
By assumption $\epsilon_1 \perp X_1$
$Z_1, Z_2, Z_3, Z_4$ i.i.d.$\Rightarrow$ $X_1 \perp X_2, X_3, X_4$
Hence,
$$
f_{\epsilon_1, X_1, X_2, X_3, X_4}=f_{\epsilon_1, X_1} f_{X_2, X_3, X_4}=f_{\epsilon_1} f_{X_1} f_{X_2, X_3, X_4}= f_{\epsilon_1}f_{X_1,X_2, X_3, X_4}
$$
repeat for $i=2,3,4$
(2017.09.16) Is it me or is the plague of silent revenge downvotes spreading? Yet one more...


You might recognize that your question is a special case of the following:


  Let $(U,V,W)$ denote random variables such that $U$ is independent of $V$, and $(U,V)$ is independent of $W$, then $U$ is independent of $(V,W)$.


This seems direct, for example computing the characteristic function $$\varphi_{U,V,W}(u,v,w)=E[\exp(iuU+ivV+iwW)]$$
(2017.09.16) Is it me or is the plague of silent revenge downvotes spreading? Yet one more...


You might recognize that your question is a special case of the following:


  Let $(U,V,W)$ denote random variables such that $U$ is independent of $V$, and $(U,V)$ is independent of $W$, then $U$ is independent of $(V,W)$.


This seems direct, for example computing the characteristic function $$\varphi_{U,V,W}(u,v,w)=E[\exp(iuU+ivV+iwW)]$$