# Large-sample distribution of the Autocovariance function

I'm trying to understand property 1.1 below ($x_t$ is white noise), from Theorem A.7(also below).

From the formula for the $W$ matrix, using the more convenient one, I get $W_{pq}=0$ when $p\neq q$ and for $p=q$, $W_{pq}=\rho_x(0)^2=\sigma_x^4$. However, in property 1.1, I get $W_{pq}=1$.

So, where have I gone wrong?

Any help would be appreciated.

This is the problem... In property 1.1 we're dealing with the asymptotic distribution of the autoCORRELATION, not autocovariance. And the correlation of white noise is 1 for $p=q$ and 0 otherwise, which explains the asymptotic covariance of the autocorrelation estimator.
In theorem A.7, $\sigma_{\hat{p_x}(h)}^2 = \frac{W}{n}$ then for $n$ large, $\sigma_{\hat{p_x}(h)}^2 = \frac{W}{n} \approx \frac{1}{n}$
based on central limit theorem result $\sqrt{n} \; \hat{p_x} \xrightarrow{d} N(0,1)$, here's for details.