I am working on the proof of $n^{1/2}[\hat{\rho}(1),...,\hat{\rho}(h)]'\xrightarrow{d}(z_1,...,z_h)'$, where $\hat{\rho}(h)$ is the sample ACF of $x_t$, $\{x_t|t=...,-2,-1,0,1,2,...\}$ are i.i.d.$(0,\sigma^2)$, and $z_t$ i.i.d.~$N(0,1)$.
I have already shown that $n^{-1}\sum_{t=1}^n x_t^2\xrightarrow{p}\sigma^2$ by WLLN. Now, I and stuck with proving $1/n\sum_{t=1}^{n-h}(x_t-\bar{x})(x_{t+h}-\bar{x})\xrightarrow{p}\sigma^2$. After proving that, I can use Martingale central limit theorem to prove $n^{1/2}\hat{\rho}(h)\xrightarrow{d}z_h$ and then use Wold device to finish the proof.
Now I am stuck with convergence of covariance, though I might have worked out some way to show it. Because $(x_t-\bar{x})(x_{t+h}-\bar{x})$ should be martingale difference sequence, we just need to use Weak Law of Large Number to show $\frac{1}{n}\sum_{t=1}^n(Y_t^2-E(Y_t^2|F_{t-1}))\xrightarrow{p}\sigma^2$, where $Y_t=(X_t-\bar{X})(X_{t+h}-\bar{X})$. But I am stuck here for a long while, so can someone please give some help or point some other method?
Appreciate that.
Note, the problem is from Shumway Time Series Analysis and Its Applications With R Examples, 4th Edition, Chapter 1, exercise 1.32, question (d).
