I'm trying to figure out the role of interleaving in communication. All I found in the Internet are some high-level interpretations of it, e.g. spreading out the error to against burst errors. I also found in a paper saying for a channel with additive ergodic noise, one can use interleaving argument and treat the channel as with additive i.i.d. noise.
I'm looking for a more mathematical proof for this 'interleaving produce uncorrelated noise' argument, and I guess this uncorrelated (or i.i.d., I'm not sure) comes from the random selection of the permutation in interleaving. I tried to prove that $P(U_{\Pi}(i)=u|U_{\Pi}(j)=v)=P(U_{\Pi}(j)=v|U_{\Pi}(i)=u)$, which is the condition for independent random varialbes. Take the random selection of permutation in consideration, $$\begin{align} P(U_{\Pi}(i)=u|U_{\Pi}(j)=v)=\sum_{\pi}P(U_{\Pi}(i)=u|U_{\Pi}(j)=v,\Pi=\pi)Pr(\Pi=\pi). \end{align}$$ I have no clue how to continue.