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Central Limit Theorem for correlated sequences - A simple proof?

This page sketches some ideas for a proof of a Central Limit Theorem for correlated sequences. This is preliminary and exploratory.

Let [eq1] be a sequence of random variables and let Xbar_n be the mean of the first n terms:[eq2]

Lindeberg-Lévy CLT states that:[eq3]

under the assumptions that:

  1. [eq1] is an independent and identically distributed (IID) sequence;

  2. the first two moments are finite[eq5]

In the Lindeberg-Lévy CLT the sequence [eq6] is required to be IID. In Central Limit Theorems for correlated sequences this requirement is usually weakened by requiring [eq7] to be stationary and mixing. These theorems state that:[eq8]where[eq9]under the assumptions that:

  1. [eq1] is stationary and mixing;

  2. the first two moments are finite[eq5]

  3. other technical assumptions are satisfied. These additional technical assumptions are usually quite cumbersome (see e.g. the Wikipedia article on the CLT or Durrett, R. (2010) "Probability: Theory and Examples", Cambridge University Press; White, H. (2001) "Asymptotic theory for econometricians", Academic Press)

My question is: are these further (and cumbersome) technical assumptions really needed? I am thinking of a proof that does not require them. I am sketching it below.

Define a new sequence $overline{Y}_{n}$ as follows:[eq12]where:[eq13]

Obviously, [eq14] and [eq15] have the same limit in distribution (because they are the same sequence!).

Now, consider the sequence $overline{Z}_{n}$ defined as follows:[eq16]

It seems natural that also [eq14] and [eq18] have the same limit in distribution (dropping the $b_{n}$ terms is like dropping a one-dimensional segment when you are integrating over a rectangle - dropping a set of measure zero does not change the results; actually, the above sum should be easily embeddable in an integral). Now, as n tends to infinity, $overline{Z}_{n}$ is a sum of IID random variables (by ergodicity and mixing). Therefore, Lindeberg-Lévy CLT applies to $overline{Z}_{n}$ and the proposition is proved (really?).

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