I am looking for proving the following variances:

a) $ var[\textbf{f}\centerdot (\textbf{f}\ast \textbf{g})] = N ~var(\textbf{X}^{2}\textbf{Y}) + (\sum_{i=(N-1)/2}^{N-1} i+ \sum_{i=(N-1)/2}^{N-2} i~)~var(\textbf{X} \textbf{Y} \textbf{Z}) + 2\binom{N}{2} ~cov(\textbf{X}^{2} \textbf{Y}, \textbf{Z}^{2} \textbf{Y})$

b) $ var[\textbf{h}\centerdot (\textbf{f}\ast \textbf{g})]= (N^{2}- 2 \sum_{i=1}^{(N-2)/2} i~)~var(\textbf{X} \textbf{Y} \textbf{Z}))$

where $ \textbf{f}= (…,0,0,f_{-(N-1)/2},…, f_{-1},f_{0},f_{1},…,f_{(N-1)/2},0,0,…)$ , $ \textbf{g}= (…,0,0,g_{-(N-1)/2},…, g_{-1},g_{0},g_{1},…,g_{(N-1)/2},0,0,…)$ , and $ \textbf{h}= (…,0,0,h_{-(N-1)/2},…, h_{-1},h_{0},h_{1},…,h_{(N-1)/2},0,0,…)$ . $ \textbf{X}, \textbf{Y}, \textbf{Z}$ are independent and identically normal distributions with mean zero and variance $ \sigma^{2}$ .

$ \ast$ denotes convolution, $ \centerdot$ denotes dot product.

Anyone can help me?

Thanks