Let $X_n$ be a sequence of identically distributed (e.g., binomial $B(1,1/2)$) random variables which are not independent (say, for any $n$ and $m$, $corr(X_n,X_m)=c$).
What can be said about the limit of their mean $\sum_n X_n/N$?
Motivation: a reason price variations assumed to be Gaussian is that they are composed by allegedly independent actions of many traders. However, in reality the traders are not independent, because, e.g., they all read the same news. So, $X_n$ is best viewed as a "set", not an "ordered sequence".