Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} e^{-\frac{\rho ^2}{4}} \rho \sum _{k=1}^{\infty } \cos (2 k x ) \cos (2 k \psi )\left(I_{k+\frac{1}{2}}\left(\frac{\rho ^2}{4}\right)+I_{k-\frac{1}{2}}\left(\frac{\rho ^2}{4}\right)\right) $$ where $\rho >0, 0<\psi<\frac{\pi}{2}, 0<x<\frac{\pi}{2}$ and $I_k(x)$ - modified Bessel function of the first kind and the summation is over all odd indices.
The thing is that the series converges slowly (because of the cosine terms). So it is hard to use this representation in practice. At first I tried to sum up the series but at last gave up. But I know that in similar problems the pdf is usually assumed to be the Von Mises or wrapped normal.
So at first I noticed that that the factor before the sum compensates the increment of the Bessel functions $\lim_{\rho \to \infty }\sqrt{\frac{2}{\pi }} e^{-\frac{\rho ^2}{4}} \rho \left(I_{k+\frac{1}{2}}\left(\frac{\rho ^2}{4}\right)+I_{k-\frac{1}{2}}\left(\frac{\rho ^2}{4}\right)\right) =\frac{4}{\pi }$. And then, manipulating the Von Mises pdf obtained: $$f_1(x,\rho,\psi)=\frac{e^{\frac{1}{4} \rho^2 \cos (2 (x -\psi ))}+e^{\frac{1}{4} \rho^2 \cos (2 (x +\psi ))}}{\pi I_0\left(\frac{\rho ^2}{4}\right)}$$

The same with wrapped normal.

$$f_2(x,\rho,\psi)=\frac{\vartheta _3\left(x -\psi ,e^{-\frac{2}{\rho ^2}}\right)+\vartheta _3\left(x +\psi ,e^{-\frac{2}{\rho ^2}}\right)}{\pi }$$ where $\vartheta _3$ is the Jacobi theta function. Well, but this is just "manipulating". May be you can give me a hint how to show it analytically?
After that I tried to compare those approximations. There are a lot of criteria to define how close those distributions are, and they all come down to computational procedure with different parameter $\rho, \psi$, which is not vary nice/descriptive. What I really want to find is a strict enough proof that this or that approximation (in analytic form) is superior for different $\rho, \psi$ in application to those pdf's. Do you think it is possible?


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