On Numerical Estimation of Joint Probability Distribution from Lebesgue Integral Quadratures

An important application of Lebesgue integral quadrature[1] is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$-- and $g$-- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density--matrix correlation. The examples of the density--matrix correlation can be the value--correlation $V_{f_i;g_j}$, similar to the regular correlation concept, and a new one, the probability--correlation $P_{f_i;g_j}$. The theory is implemented numerically; the software is available under the GPLv3 license.