Power Spectral Density

The sufficient statistic for model two is the periodogram ${\displaystyle I(j/n)}$. The mean is ${\displaystyle \frac{2{\gamma }_j^2}{n}}$, this is the power of frequency ${\displaystyle \frac{j}{n}}$.
If we plot the points ${\displaystyle (\frac{j}{n},f\left(\frac{j}{n}\right))}$ for ${\displaystyle j=1,\cdots ,m}$ and join the neighboring points by lines, we get a continuous function plot. This is known as the power spectral density and is defined on ${\displaystyle [0,0.5]}$.
After estimating ${\displaystyle {\gamma }_1^2,\cdots ,{\gamma }_m^2}$, it is customary to plot ${\displaystyle f}$.

Equivalent Definition: Rewriting the Model in Terms of ${\displaystyle {\displaystyle y_t}}$

Model three is written in terms of ${\displaystyle b_j}$: $$\mathrm{Re}(b_j),\mathrm{Im}(b_j)\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}}{\sim }\mathcal{N}(0,{\gamma }_j^2).$$
Recall inverse DFT: $$y_t=\frac{1}{n}\sum _{j=0}^{n-1}{b}_j\exp \left(\frac{2\pi ijt}{n}\right).$$ Rewrite it into $$\begin{array}{rl}{y}_t& =\frac{1}{n}\sum _{j=0}^{n-1}(\mathrm{Re}(b_j)+\mathrm{i}\mathrm{Im}(b_j))(\cos \left(\frac{2\pi jt}{n}\right)+\mathrm{i}\sin \left(\frac{2\pi jt}{n}\right))\\ & =\frac{1}{n}\sum _{j=0}^{n-1}(\mathrm{Re}(b_j)\cos \left(\frac{2\pi jt}{n}\right)-\mathrm{Im}(b_j)\sin \left(\frac{2\pi jt}{n}\right))\\ & +\frac{\mathrm{i}}{n}\sum _{j=0}^{n-1}(\mathrm{Re}(b_j)\sin \left(\frac{2\pi jt}{n}\right)+\mathrm{Im}(b_j)\cos \left(\frac{2\pi jt}{n}\right)).\end{array}$$
Ignore the imaginary part: $$\begin{array}{rl}{y}_t& =\frac{1}{n}\sum _{j=0}^{n-1}(\mathrm{Re}(b_j)\cos \left(\frac{2\pi jt}{n}\right)-\mathrm{Im}(b_j)\sin \left(\frac{2\pi jt}{n}\right))\\ & =\frac{b_0}{n}+\frac{1}{n}\sum _{j=1}^{n-1}(\mathrm{Re}(b_j)\cos \left(\frac{2\pi jt}{n}\right)-\mathrm{Im}(b_j)\sin \left(\frac{2\pi jt}{n}\right)).\end{array}$$
Recall we assume ${\displaystyle n}$ is odd and ${\displaystyle m=\frac{n-1}{2}}$. Also use the fact from here that ${\displaystyle b_{n-j}={\bar{b}}_j}$, which is ${\displaystyle \mathrm{Re}(b_{n-j})=\mathrm{Re}(b_j)}$ and ${\displaystyle \mathrm{Im}(b_{n-j})=-\mathrm{Im}(b_j)}$. This gives $$y_t=\frac{b_0}{n}+\sum _{j=1}^m(\frac{2\mathrm{Re}(b_j)}{n}\cos \left(\frac{2\pi jt}{n}\right)-\frac{2\mathrm{Im}(b_j)}{n}\sin \left(\frac{2\pi jt}{n}\right)).$$
Denote ${\displaystyle {\beta }_0=\frac{b_0}{n},{\beta }_{1j}=\frac{2\mathrm{Re}(b_j)}{n},{\beta }_{2j}=-\frac{2\mathrm{Im}(b_j)}{n}}$, then $$\begin{array}{}\text{(2.1)}& y_t={\beta }_0+\sum _{j=1}^m({\beta }_{1j}\cos \left(\frac{2\pi jt}{n}\right)+{\beta }_{2j}\sin \left(\frac{2\pi jt}{n}\right)).\end{array}$$
So model three is equivalent to (2.1), with $${\beta }_{1j}=\frac{2\mathrm{Re}(b_j)}{n}\sim \mathcal{N}(0,\frac{4}{n^2}{\gamma }_j^2),{\beta }_{2j}=-\frac{2\mathrm{Re}(b_j)}{n}\sim \mathcal{N}(0,\frac{4}{n^2}{\gamma }_j^2).$$

Two Key Properties of the Spectrum Model

Consider definition 2 (2.1) of the spectrum model.
First is (recall here) $$\mathrm{Var}(y_t)=\sum _{j=1}^m{\tau }_j^2=\frac{2}{n}\sum _{j=1}^{m}f\left(\frac{j}{n}\right)\approx 2{\int }_0^{\frac{1}{2}}f(w)\mathrm{d}w.$$
Next is $$\begin{array}{rl}\mathrm{Cov}(y_t,y_{t+h})& =\sum _{j=1}^m{\tau }_j^2\cos \left(\frac{2\pi jh}{n}\right)=\frac{2}{n}\sum _{j=1}^{m}f\left(\frac{j}{n}\right)\cos \left(\frac{2\pi jh}{n}\right)\\ & \approx 2{\int }_0^{\frac{1}{2}}f(w)\cos (2\pi wh)\mathrm{d}w.\end{array}$$

The Case of Even ${\displaystyle {\displaystyle n}}$

If ${\displaystyle n}$ is even, ${\displaystyle \frac{1}{2}}$ becomes a Fourier frequency and ${\displaystyle b_{\frac{n}{2}}}$ becoms real (${\displaystyle \sin (\pi t)=0}$). Now we can simple avoid working with ${\displaystyle \frac{1}{2}}$ by taking ${\displaystyle m=\frac{n-2}{2}}$ and using $$\mathrm{Re}(b_j),\mathrm{Im}(b_j)\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}}{\sim }\mathcal{N}(0,{\gamma }_j^2).$$ This will be equivalent to (2.1).