Central Limit Theorem and Convergence

Question

I'm trying to approach the following question:

Let $\{X_i\}_{i=1} ^\infty$ be a sequence of identically distributed random variables, with $\mathbb{E}(X_i)=0$ and finite variance $\sigma^2$.

For all $i,j\in\mathbb{N}$ s.t. $i+2\le j$, $X_i,X_j$ are independent.

Let $Z\sim N(0,\sigma^2)$.

I'm trying to understand why can't I use the CLT here to say that by separating $\{X_i\}_{i=1} ^\infty$ to $\{X_{2i}\}_{i=1} ^\infty$ and $\{X_{2i+1}\}_{i=1} ^\infty$, we can conclude that

$\frac{1}{\sqrt n}\sum_{i=1}^{n}X_i\overset{\text d}{\to}Z$

Answer 1

What you say can be true, for example when the $X_i$ are all independent and identically distributed. But it does not need to be true.

Let's suppose as a counter-example:

• we have sequences of $Y_i$ and $W_i$ all i.i.d. $N(0,\sigma^2)$
• $X_{2j}=(Y_j+W_j)/\sqrt{2} \sim N(0,\sigma^2)$
• $X_{2j+1}=(Y_j+W_{j+1})/\sqrt{2} \sim N(0,\sigma^2)$
• and each $X_j$ is independent of the others which are not its immediate neighbour

Then $\frac{1}{\sqrt n}\sum_{i=1}^{n}X_i \approx \frac{1}{\sqrt n}\sum_{j=1}^{n/2} 2 X_{2j} \overset{\text d}{\to} N(0,2\sigma^2)$ rather than $N(0,\sigma^2)$ as $n$ increases