Let $X$ and $Y$ be hyperbolically distributed, and let $f_{X} (x)$ and $f_{Y} (x)$ be their associated density functions. Then $$f_{X} (x) = \frac{\gamma}{2 \alpha \delta K_{1} ( \delta \gamma) } e^{- \alpha \sqrt{\delta^{2}+(x-\mu)^{2}} + \beta (x - \mu)} , $$ where $K_{1} ( \cdot ) $ is a Bessel function of the second kind (link), $\alpha, \beta, \delta \in \mathbb{R}$ and $\gamma = \sqrt{\alpha^{2}-\beta^{2}}$. Let's denote this by saying $X \sim H( \alpha, \beta, \delta, \mu)$. You can find more information on the hyperbolic distribution over here.

I am wondering how to compute the convolution $(f_{X} * f_{Y})(x)$ . Whether it is done directly or perhaps by means of Laplace/Fourier transformations and the convolution theorem. Do you have a source addressing this problem? Or do you know how to compute this convolution yourself?

If it is not possible to compute the convolution analytically, what would be a good way to approximate it?

Thank you in advance!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.