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Is there any version of CLT does not need independent $X_i$'s? The reason I'm asking this is because I'm woking on a moving average problem. I have 3 pieces each by CLT converges in distribution to standard normal. However, when I combine the three pieces, they have some terms in common and are not independent?

Thanks

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There are plenty of such theorem. See for instance CLTs for mixing processes and Doukhan's book "Mixing: Properties and Example".

See also chapter 7 in the book "Weak Dependence: with examples and applications".