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 be a sequence of random variables and let be the mean of the first terms:

Lindeberg-Lévy CLT states that:

under the assumptions that:

is an independent and identically distributed (IID) sequence; the first two moments are finite

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

is stationary and mixing; the first two moments are finite 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 as follows: where:

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

Now, consider the sequence defined as follows:

It seems natural that also and have the same limit in distribution (dropping the 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 tends to infinity, is a sum of IID random variables (by ergodicity and mixing). Therefore, Lindeberg-Lévy CLT applies to and the proposition is proved (really?).