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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 $m$, $corr(X_n,X_m)=c>0$).

What can be said about the limit of their mean $\frac{1}{N}\sum_n X_n$?

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.

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  • $\begingroup$ What kind of limit interests you? $\endgroup$
    – Davide Giraudo
    Nov 7, 2012 at 20:51
  • $\begingroup$ any limit is fine $\endgroup$
    – sds
    Nov 8, 2012 at 16:14
  • $\begingroup$ Actually, it's more related to law of large numbers than central limit theorem. $\endgroup$
    – Davide Giraudo
    Nov 8, 2012 at 16:20
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    $\begingroup$ Do you need limit laws for exchangeable random variables? $\endgroup$
    – tchakravarty
    Nov 8, 2012 at 17:15

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I think little or nothing can be said with no more assumptions than stated. With more assumptions about the dependence among the $X_n$, some limit theorems are proved in this paper.

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