I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& \frac{1}{2\pi}\sum_{\omega\in\left[0,2\pi\right]:f\left(\omega\right)=\lambda} \frac{1}{\left|f'\left(\omega\right)\right|} \tag{1a} \\ \overset{\text{?}}{=}{}& \frac{1}{\pi(\vartheta+\varphi\lambda)\sqrt{\left[(1+\vartheta)^{2}-\lambda(1-\varphi)^{2}\right]\left[\lambda(1+\varphi)^{2}-(1-\vartheta)^{2}\right]}}\mathbf{1}_{(\lambda_{-},\lambda_{+})}(\lambda) \text{,} \tag{1b} \end{align} with \begin{equation} \lambda,\vartheta,\varphi\in\mathbb{R}\text{,}\;\left|\varphi\right|<1\text{,} \quad \lambda_{-} = \min{(\lambda^{-},\lambda^{+})}\text{,} \quad \lambda_{+} = \max{(\lambda^{-},\lambda^{+})}\text{,} \quad \lambda^{\pm} = \frac{(1\pm\vartheta)^{2}}{(1\mp\varphi)^{2}} \text{.} \end{equation} $f$ is the Fourier transform of the autocovariance function of a $\operatorname{ARMA}\left(1,1\right)$ process with $\operatorname{MA}\left(1\right)$ polynomial $a$ and $\operatorname{AR}\left(1\right)$ polynomial $b$. It is given by \begin{equation} f\left(\omega\right) = \left|\frac{b\left(\operatorname{e}^{\mathsf{i}\omega}\right)}{a\left(\operatorname{e}^{\mathsf{i}\omega}\right)}\right|^{2} = \frac{1+\vartheta^{2}+2\vartheta\cos\left(\omega\right)}{1+\varphi^{2}-2\varphi\cos\left(\omega\right)} \text{,} \tag{2} \end{equation} for $\omega\in\left[0,2\pi\right]$. The derivative of $f$ is given by \begin{equation} f'\left(\omega\right) = -\frac{2\left(\varphi+\vartheta\right)\left(1+\varphi\vartheta\right)\sin\left(\omega\right)}{\left(1+\varphi^{2}-2 \varphi \cos (\omega )\right)^2} \text{.} \tag{3} \end{equation}

Any ideas on how to solve equation (1)? The solution looks as if it was calculated by use of the residue theorem, but I do not know how to start (or how to transform the implicit sum to an integral).

Any help is much appreciated.

  • $\begingroup$ it reminds me at the identify $\delta(f(x))=\sum_{x_i,f(x_i)=0} \delta(x-x_i)$ but i'm not 100% sure if this is the right direction $\endgroup$ – tired Aug 21 '15 at 16:41

I just visited this problem again and made the effort to calculate the function $g\left(\lambda\right)$ by hand.

As a result, I noticed that the authors omited a factor $\left(\vartheta+\varphi\right)\left(1+\vartheta\varphi\right)$ in equation (1b). The following equation (3.2) in their paper is right again, so it seems the authors simply forgot the mentioned factor.


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