Let $Y=(X_1,X_2)$ be a random vector such that $X_1$ and $X_2$ are iid according to $f_X$: $$f_{Y}(x_1,x_2)= f_1(x_1)f_2(x_2)=\prod_{i=1}^2 f_X(x_i)$$
Consider $Y^1= (X_1^1 , X_2^1)$, $Y^2= (X_1^2 , X_2^2)$, $Y^3= (X_1^3 , X_2^3)$ ... an infinite sequence of random vectors iid according to $Y=(X_1,X_2)$. Now, suppose $N$ a random natural number independent of the sequence $(Y^j)_{j=0}^\infty$. Define $$Z=Y^1 + Y^2 + ...+ Y^N$$ Note that $Z=(Z_1,Z_2)$, where $Z_i=X_i^1+X_i^2+...+ X_i^N$ for $i=1,2$. Suppose that $N\sim \hbox{Poisson}(1)$. My question is: how to show that $Z_1$ and $Z_2$ are iid?
It is easy to show that $Z_1$ and $Z_2$ have the same distribution, by conditioning on $N$. But $Z_1$ and $Z_2$ are not independent, since the value of $Z_1$ can give you information about the value of $N$, and therefore can give you knowledge about the value of $Z_2$. (Imagine the case where $N$ takes two possible values, either very small or very large.) However, $Z_1$ and $Z_2$ are conditionally independent given $N$.