I have a non-homogeneous Poisson point process $X = (x_k, y_k)$ with intensity parameter $\lambda(x,y)$. For any two random points $(x_i,y_i)$ and $(x_j, y_j)$, I am forming the (product) random variables, $R_1 = x_ix_j+y_iy_j$, basically a dot product, and $R_2 = x_iy_j-x_jy_i$. I am trying to understand the distribution of $R1$ and $R2$.
In the case of homogenous Poisson process with a constant intensity $\lambda$, I get the following density plot for $R1$:
This resonates with the following post that suggests approximating the PDF of the product of normal random variables by modified Bessel function of the second kind. If $\lambda$ is sufficiently large, then homogenous Poisson is approximately normal and so this kinda makes sense. But for small $\lambda$ or in the case of non-homogenous Poisson, I am not sure if there is an analytic formula (or distribution) for the PDF of $R1$ and $R2$.
Disclaimer: I am not a mathematician/statistician but a biologist trying to learn some stats through programming. Thanks!