We know that the lag window estimator of the spectral density is defined as: $$S^{(lw)}(f)=\int_{-f_{(N)}}^{f_{(N)}}W_m(f-\phi)\hat{S}^{p}(\phi)d\phi=\sum\limits_{\tau=-(N-1)}^{\tau=N-1}w_{\tau,m}s^{(p)}_{\tau}e^{-2\pi f\tau}$$ Where $W_m(f)$ (the spectral window) is the Fourier transform of $w_{\tau,m}$( the lag window), and $\hat{S}^{p}$ (The periodogram ) is the Fourier transform of $s^{(p)}_{\tau}$ (the covariance function).
The covariance between $S^{(lw)}(f)$ and $S^{(lw)}(f')$ is approximatly $0$ if $|f-f'|$ is larger than the width (or bandwidth) of the lag window $W_m$. But if we write the following: $$cov\{S^{(lw)}(f),S^{(lw)}(f')\}=E\{S^{(lw)}(f)S^{(lw)}(f')\}- S(f)S(f')$$ $S(f)$ being the true spectral density at frequency $f$. We can write then: $$E\{S^{(lw)}(f)S^{(lw)}(f')\} =\frac{1}{N}\sum\limits_ {f,f'}W_m(f)W_m(f')E\{\hat{S}^{(p)}(f)\hat{S}^{(p)}(f')\}$$ $$=\frac{1}{N}\sum\limits_ {f,f'}W_m(f)W_m(f')E\{\hat{S}^{(p)}(f)\}E\{\hat{S}^{(p)}(f')\}=S(f)S(f')$$ because we know that the periodogram ordinates are uncorrelated for $|f-f'|> \frac{1}{N}$. From what I have just written the ordinates of the lag window estimator are uncorrelated onlu if the peridodogram ordinates are uncorrelated ($|f-f'|> \frac{1}{N}$), which is inconsistent with the result that we already know. Where am I mistaken?
