I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the random variables be dependent for instance?
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1$\begingroup$ You may take a look at Stochastic Limit Theory: An Introduction for Econometricicans by James Davidson. He discusses lots of ways to weaken independence assumptions in the CLT and also how to turn these one-dimensional results into multivariate ones. $\endgroup$– Michael GreineckerCommented Nov 6, 2013 at 16:19
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1$\begingroup$ @Davide I think he refers to the problem of showing that an appropriatly rescaled sequence of $n$-dimensional random vectors has asymptotically a multivariate normal distribution. $\endgroup$– Michael GreineckerCommented Nov 6, 2013 at 16:20
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$\begingroup$ Indeed Michael, you have interpreted my question correctly, thx. $\endgroup$– wanderfloCommented Nov 6, 2013 at 16:34
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$\begingroup$ In the future, this question might be better received at the Stats version of MathOverflow. $\endgroup$– David WhiteCommented Nov 6, 2013 at 16:42
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1$\begingroup$ The site is stats.stackoverflow.com. It's probably true that this question would be better received there, but the reason is not that it's statistics and not mathematics. It's that they're more welcoming of reference-request questions. $\endgroup$– Mark MeckesCommented Nov 6, 2013 at 19:36
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1 Answer
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The classics:
Hall and Heyde's book http://www.amazon.com/Martingale-Application-Probability-Mathematical-Statistics/dp/0123193508
Bhattacharya-Rao's book http://books.google.co.il/books/about/Normal_Approximation_and_Asymptotic_Expa.html?id=H1lOIVHcRDEC&redir_esc=y
answered Nov 6, 2013 at 18:25