I'm trying to understand Nason et al(2000)'s proof of Proposition 3.3 in this paper: http://www.maths.bris.ac.uk/~guy/Research/papers/WavProcEWS1.pdf
The expectation of the uncorrected periodogram is given, and I don't fully understand the step after the substitution of $m=n+k$, where the use of property (10) and (12) gives $(w_{l,n+k;T})^2= S(\frac{n+k}{T})+O(T^{-1})$. I believe this is because the approximation to $|W_j(z)|^2$ is just $S(j)$ by property (12) and then property (10) gives that $W_j(\frac{k}{T})$ and $w_{j,k;T}$ only differ by a max of $\frac{C_j}{T}$. Thus they are saying $w_{l,n+k;T}^2=(W_l(\frac{n+k}{T})-W_l(\frac{n+k}{T})+w_{l,n+k,T})^2=[S(\frac{n+k}{T})+O(\frac{1}{T})]$? I am not sure at this point where the squared term disappears off to or if my interpretation is correct even up to this point.
Then assuming their line, I don't fully understand what the Lipschitz property has to do with the next equality? Can someone help me understand this?
Following the expansion of the terms I can follow the rest of the proof, as they are just the definitions provided by Nason et al(2000), but these steps are beyond my understanding. Thanks in advance!