  Statistical independence is a notion ubiquitous in various fields such as in
statistics, probability, number theory and physics. We establish the stability
of independence for any pair of random variables by their corresponding
Brockwell transforms (Brockwell, 2007) beyond the non-atomic condition that is
naturally imposed on their distributions, thereby generalizing the proposition
originated by Cai et al. (2022). A central novelty in our work is to formulate
the problem as a possibly new type of mathematical inverse problem, which aims
to claim that an integral equation has a unique solution in terms of its
stochastic kernel -- not under-determined contrary to the usual cases, and also
to design a recursive constructive scheme, combined with the property of the
quantile function, that solves the aforementioned integral equation in an
iterative manner.
