Given the bivariate Cauchy distribution: $$f(x,y; x_0, y_0) \sim \frac{1}{2\pi}\frac{1}{((x - x_0)^2 + (y - y_0)^2+1)^{1.5}}$$ How do you generate samples appropriately?
I am aware of inverse transforms utilizing the CDF definition, however I can only find articles pertaining to the single variable case.
Thanks!
Attempt:
Let $\mathbf{Y}$ be a random vecor of $D$ components
- Generate $D-1$ random variables $v_1, v_2, \ldots, v_{D-1}$ of univariate Cauchy distribution via $v_i = \text{tan}(\pi(U - 1/2))$ where $U$ is a uniform on $[0,1]$
- Generate $w_2, w_3, \dots, w_D$ random variables of a Student t-distribution with $m = 2,3, \dots, D$ degrees of freedom.
- Set $x_1 = v_1$
- Iterate the following for $x_m,$ $m = 2,3,\ldots, D$ $$x_m = m^{-1/2}\left(1+\sum_{i=1}^{m-1}v_i^2\right)^{1/2}w_m$$
So in the bivariate case, $x_1 = v_1$ and $$x_2 = 2^{-1/2}\left(1+v_1^2\right)^{1/2}w_2$$
and $\mathbf{Y} = \mathbf{X} + \mathbf{Y}_{\text{curr}}$ since the given scale parameter is 1?
Is this the correct approach?
A multivariate Cauchy distribution coincides with a multivariate Student's t distribution with 1 degree of freedom. You can generate the Student's t distribution samples via Normals with standard sample-generating theory and Cholesky. Stay safe