Given the arrays $C=[C_1,C_2,...,C_N]$ and $S=[S_1,S_2,...,S_N]$ of lengths $N$ with elements that are discrete iid uniform distributed with equal probability (p=1/2) of being $\pm$ 1
Consider the random variables (for a given $l, n, m$):
$W=C_lC_mC_n$
$X=S_lS_mC_n$
$Y=C_lS_mS_n$
$Z=S_lC_mS_n$
It can be shown that these random variables ($W, X, Y, Z$) are zero mean, uniform distributed with equal probability (p=1/2) of being $\pm$ 1. Furthermore, it can be shown that they are uncorrelated (e.g. $E[WX]=E[(C_lC_mC_n)(S_lS_mC_n)]=0$ since they are zero mean ($E[C_i]=E[S_i]=0$) and $C_i^2=S_i^2=1$ ).
Now how can one go about showing that the random variables ($W, X, Y, Z$) are pairwise independent? (The expectation $E[WX]=E[(C_lC_mC_n)(S_lS_mC_n)]=0=E[W]E[X]=0$ but that doesn't necessarily mean they are independent). So I'm wondering what is a good way to show independence, any tricks one can use?