A central limit theorem for dependent random variable.

Question

Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e. $$ u_j\stackrel{d}{=}\text{N}(0,1). $$ Call $\mu_r=\mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution of the random variable $$ \sqrt{n}\,\left(\frac{1}{n}\sum_{j=1}^{n}u_j^2-\frac{\mu_1^{-2}}{n}\sum_{j=1}^{n}|u_j|\,|u_{j+1}|\right)=n^{-1/2}\,\sum_{j=1}^n|u_j|\left(|u_j|-\mu_1^{-2}\,|u_{j+1}|\right)=n^{-1/2}\,\sum_{j=1}^n\xi_j, $$ where $\xi_j=|u_j|\left(|u_j|-\mu_1^{-2}\,|u_{j+1}|\right)$. However, since the $\xi_j$ are dependent, I do not know how to proceed.

Answer 1

Since $(\xi_j)_{j\le N}$ and $(\xi_j)_{j\ge N+k}$ are independent for $k\ge 2$, the stochastic process $(\xi_j)_{j\in \mathbb{N}}$ is strongly($\alpha$-) mixing. More precisely, we can say that the strong mixing coefficient $\alpha(k)$ vanishes for $k\geq 2$. Note that $E\xi_j =0$ and every $p$-th moment $E|\xi_j|^p$ is finite. By $\alpha$-mixing central limit theorem, we have $$ \frac{1}{\sqrt{n}}\sum_{j=1}^n \xi_j \to_d \mathcal{N}(0,\sigma^2) $$ where $\sigma^2 = E[\xi_j^2] +2 E[\xi_j \xi_{j+1}]\ge 0$.