Does a sequence of random variables constructed in a certain manner converge in distribution to a Gaussian? Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of of IID random variables taken for simplicity with mean zero and variance one.
The Central Limit Theorem give us that 
$$
\frac{X_1 + \dots + X_n}{\sqrt{n}} \xrightarrow {d} N\left(0,1\right)
$$
If one constructs a new sequence $\{Y_n\}_{n \in \mathbb{N}}$ from the first one given by the square of the sum of two consecutive terms, i.e.
$$Y_1 = (X_1 + X_2)^2, Y_2 = (X_2 + X_3)^2, \dots , Y_n = (X_n + X_{n+1})^2 $$
do there exist two sequences $\{\mu_n\}_{n \in \mathbb{N}}$ and $\{\sigma_n\}_{n \in \mathbb{N}}$ s.t. 
$$\frac{1}{\sigma_i^2} \sum_{i=1}^n ( Y_i - \mu_i) \rightarrow N(0,1) $$
I was thinking of using some Lyapunov type central limit theorem to prove this but there is an obvious (weak) dependence in the sequence. Is it possible to show this or is it not true?
 A: Let $\left(X_i\right)_{i\geqslant 1}$ be an i.i.d. sequence and let $f\colon \mathbb R^2\to \mathbb R$ be a function such that such that the random variable $Y_i:=f(X_i,X_{i+1})$ is centered and square integrable.
Let $n$ be a fixed integer and $q\in\left\{1,\dots,n\right\}$. We write 
\begin{align}
\sum_{i=1}^nY_i&= \sum_{i=1}^{q\left\lfloor \frac nq\right\rfloor}Y_i+
\sum_{q\left\lfloor \frac nq\right\rfloor}^nY_i\\
&= \sum_{k=1}^{\left\lfloor \frac nq\right\rfloor}
\sum_{i=(k-1)q+1}^{kq}Y_i+\sum_{q\left\lfloor \frac nq\right\rfloor}^nY_i\\
&= \sum_{k=1}^{\left\lfloor \frac nq\right\rfloor}
\sum_{i=(k-1)q+2}^{kq}Y_i+\sum_{k=1}^{\left\lfloor \frac nq\right\rfloor}Y_{(k-1)q+1}+\sum_{q\left\lfloor \frac nq\right\rfloor}^nY_i.
\end{align}
Denoting 
$Z^q_k:= \sum_{i=(k-1)q+2}^{kq}Y_i$, the sequence $\left(Z^q_k\right)_{k\geqslant 1}$ is i.i.d. hence we could apply the central limit theorem but the problem is that in order to make the contribution of $n^{-1/2}\sum_{k=1}^{\left\lfloor \frac nq\right\rfloor}Y_{(k-1)q+1}$ small but we can choose $q$ depending on $n$ and apply the central limit theorem for arrays.
A: Your specification corresponds to $Y_i = f(X_{i-1},X_i)$ for $i=1,2,3,\ldots$, but CLT can be extended to more general form $Y_i = f(\cdots,X_{i-2},X_{i-1},\widehat{X_i},X_{i+1},\ldots)$ ($\widehat{X_i}$ denotes the center of the variables) given that $\ldots,Y_{i-1},Y_i$ and $Y_{i+N},Y_{i+N+1},\ldots$ are "asymptotically independent" as $N\to \infty$, which can be rigorously defined in terms of mixing conditions. The given case is among the simplest since $\ldots,Y_{i-1}, Y_i$ and $Y_{i+2},Y_{i+3},\ldots$ are independent. Proof of this fact requires a bit of knowledge in ergodic and martingale theory. If you are interested, see http://www.stat.yale.edu/~mjk56/MartingaleLimitTheoryAndItsApplication.pdf for martingale limit theory and C. C Heyde, On the central limit theorem for stationary processes, 1974 for ergodic CLT.
