I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution.
The distance of a point from the diagonal line y = x is $\frac{\sqrt{(x_0-y_0)^2}}{\sqrt{2}} \propto \text{hypotenuse of the right triangle formed by the point and the line } \sqrt{(x_0-y_0)^2}$, i.e. the range between x and y
Now, since the actual distance is distributed Lomax Pareto, and the range of |x - y| is proportional to the distance, the distribution of the range is similarly Lomax
$$ \begin{align} p(r) &= {\alpha \over \lambda} \left[{1 + {r \over \lambda}}\right]^{-(\alpha+1)}, \qquad r \geq 0 \\ &= 2 * \int_0^\infty f(z)f(z+r) dz \text{ (since there are only 2 points)} \end{align} $$
I my first attempt was to try a Lomax marginal for each variable and then work forward.
$$ \begin{align} f_{R_2}(r) &= 2 * \int_0^\infty f(z)f(z+r) dz, \qquad r \geq 0 \\ &= 2 * \int_0^\infty \frac{\alpha}{\lambda} \left\{ 1 + \frac{z}{\lambda} \right\}^{-\alpha-1} \frac{\alpha}{\lambda} \left\{ 1 + \frac{z + r}{\lambda} \right\}^{-\alpha-1} dz \\ &= 2 * \left( \frac{\alpha}{\lambda} \right)^2 \int_0^\infty \left\{ \frac{\lambda + z}{\lambda} \right\}^{-\alpha-1} \left\{ \frac{\lambda + z + r}{\lambda} \right\}^{-\alpha-1} dz \\ &= 2 \alpha^2 \lambda^{\alpha - 1} \int_0^\infty \left\{ \lambda + z \right\}^{-\alpha-1} \left\{ \lambda + z + r \right\}^{-\alpha-1} dz \\ \end{align} $$
...but I cannot push the integral through.
Are there any methods that can be used to identify one or more possible marginal distributions whose range forms a Lomax Pareto (Type II) distribution?